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## Statistics and Probability

### Fundamentals of Probability - Basic Probability

#### Q.01

'Find the probability of a draw in a rock-paper-scissors game between two players.'

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#### Q.02

'From a pile of sugar bags, 100 bags were randomly selected and weighed, resulting in an average weight of 300.4g. Assuming a population standard deviation of 7.5g, estimate the average weight per bag with 95% confidence.'

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#### Q.04

'Find the joint distribution of X and Y, and check if X and Y are independent.'

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#### Q.06

'Example 62 Binomial distribution and transformation of random variables'

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#### Q.07

'Calculate the probability of not receiving any prize in a certain event.'

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#### Q.08

'Sum, Expectation, and Variance of a Sequence\nThere are cards labeled with numbers from 1 to n, each with n, n-1, ..., 1 cards. These cards are placed into a bag. After mixing the bag well, a card is drawn, and the number on it is considered as the random variable X.\n(1) Determine the probability P(X=k).\n(2) Find E(X) and V(X) for X.'

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#### Q.09

'Calculate the probability distribution of the sum of the numbers when two dice are rolled, and find the probability of X falling within a specific range.'

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#### Q.10

'In general, if a random variable X takes on possible values x1, x2, ..., xn, with probabilities p1, p2, ..., pn respectively, what conditions must the probability P satisfy?'

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#### Q.11

"As a mathematical problem, let's consider the allocation of seats in an election using mathematical methods. Specifically, using a general election system (e.g. D'Hondt method or Sainte-Laguë method), calculate the number of seats allocated to each party based on the given number of votes. Here are the votes for each party. Please calculate the number of seats allocated to each party."

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#### Q.12

'When sampling a random sample of size 100 from a population following a normal distribution with a mean of 120 and a population standard deviation of 30, calculate the following probabilities.'

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#### Q.13

'Among n cards, each containing numbers 1,2,3,...,n, one per card. When randomly drawing 2 cards, let X be the smaller number and Y be the larger one. Note that n ≥ 2. (1) Find the probability of X=k, where k=1,2,3,...,n. (2) Find the expected value of X. (3) Find the variance of Y.'

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#### Q.14

'For a certain product A, a survey of 300 people was conducted, with 210 people supporting product A. Calculate the confidence interval at 95% confidence level for the proportion of supporters of product A in the population. Use √7=2.65 for calculation and round to the third decimal place.'

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#### Q.15

'When two red balls and one white ball are taken out of a bag, the arrangement of the balls when taking out two balls at the same time, distinguishing two red balls as red 1 and red 2 ((red 1, red 2), (red 1, white), (red 2, white)) are three possibilities. In this experiment, let the number of red balls drawn be X, then X can take values of 1 or 2, find the probabilities for X to take these values.'

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#### Q.16

'The occurrence rate of side effects of a certain drug was 4% in the past, but after using a new improved drug on 400 patients, 8 patients experienced side effects. Can we conclude that the occurrence rate of side effects has decreased? Perform a test at a significance level of 5%. Assume that the 400 patients were randomly selected.'

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#### Q.17

'The incidence rate of side effects of a certain drug was traditionally 4%, but when the improved new drug was used in 400 patients, 8 patients experienced side effects. Can it be concluded that the incidence rate of side effects has decreased? Perform a test at a significance level of 5%. Assume that the 400 patients were randomly selected.'

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#### Q.18

'Regarding the underlined part, answer the following question:\n(1) Through online transactions, it is now possible to directly purchase goods from foreign countries. However, issues such as non-delivery of goods or damages have also occurred. In such cases, the National Life Center provides consultation services. Choose one of the government agencies below that is responsible for overseeing this institution and provide the answer by number:\n1. Financial Services Agency 2. National Tax Agency\n3. Consumer Affairs Agency 4. Fair Trade Commission'

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#### Q.19

'For the statement X·Y regarding the underlined part J in question 11, choose the correct combination of true or false from the options below and answer with the corresponding number.'

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#### Q.20

'Two situations in which instructional videos are helpful'

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#### Q.21

'When tossing a coin 8 times, find the probability of getting heads 5 times or more in a row.'

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#### Q.22

'When point P is initially at the origin O on the number line, each time a die is rolled, if an even number appears, move 3 units in the positive direction, and if an odd number appears, move 2 units in the negative direction. When the die is rolled 10 times, what is the probability that point P is at the origin O?'

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#### Q.23

'Find the probability that the product of the numbers on two dice is 24 or less.'

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#### Q.24

'A random ball is taken out from a bag, and this operation continues without returning the taken ball to the bag. Find the following probabilities: (1) Probability that the red balls are taken out first (2) Probability that only 5 white balls remain in the bag after all red balls are taken out.'

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#### Q.25

'What is the probability of the sum being 10 or more when throwing two dice at the same time?'

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#### Q.26

'In boxes A, B, and C, there are red, white, and black balls respectively. The numbers are as shown in the table on the right. Pick a box randomly and take out one ball. Find the following probabilities.'

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#### Q.27

'When two dice are rolled simultaneously, what is the probability that the product of the outcomes is 24 or less?'

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#### Q.28

'Find the probability that there are 0 red balls after 3 operations.'

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#### Q.29

'Number cards from 1 to 6 are prepared with the respective quantity for each number. Which of the following options is more advantageous when drawing one card?\n(1) Receive a 100 yen coin with the same number as the drawn card.\n(2) Receive a fixed amount of 700 yen only when an even number is drawn.'

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#### Q.30

'Basic Example 47 Basic of Probability of Repeated Trials\nThere are 8 lottery tickets, including 2 winning ones. When drawing the tickets one by one with replacement for 5 times, find the following probabilities:\n(1) Probability of winning exactly twice\n(2) Probability of winning at least 4 times'

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#### Q.31

'A and B are playing a series of games, with the first to win 3 games being the overall winner. Let the probability of A winning a single game be 1/3. Answer the following question.'

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#### Q.32

'When drawing one card from the numbers 91 to 50, find the probability of drawing a card that is not a multiple of 3.'

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#### Q.33

'Throw 3 dice simultaneously. (1) Find the probability that the sum of points on any 2 of the 3 dice is 5. (2) Find the probability that the sum of points on any 2 of the 3 dice is 10. (3) Find the probability that the sum of points on any 2 dice is not a multiple of 5.'

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#### Q.34

'There are a total of 9 cards with numbers from 1 to 9 written on them, each one in one piece. When 3 cards are drawn from these, the probability that all the numbers on the cards are odd is A. Furthermore, the probability that the sum of the numbers on the 3 cards drawn is odd is B.'

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#### Q.35

'What is the probability that the minimum value is 3 or the maximum value is 4 when throwing two dice simultaneously?'

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#### Q.36

'There are 9 red number cards (1 to 9) and 6 white number cards (1 to 6) in the box. When drawing a card from this box, let event A be drawing an even card and event B be drawing a red card. Find the following probabilities.'

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#### Q.37

'Calculate the probability of rolling an even number on the first throw and a number of 4 or less on the second throw when rolling a 101-sided die.'

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#### Q.39

'When rolling two dice simultaneously, calculate the probability of both dice showing the same number, as well as the probability of the sum of two numbers being odd.'

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#### Q.40

'What is the probability that the number of red balls is at least 2 after 3 operations?'

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#### Q.41

'Since in each round a white ball is taken from bag A and a red ball from bag B, the probability to find is'

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#### Q.42

'Find the probability of getting 2 heads and 1 tail when throwing 3 coins at the same time.'

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#### Q.43

'Three people play rock-paper-scissors repeatedly. However, the person who loses cannot participate in the next round. Answer the following questions.'

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#### Q.44

'When rolling a six-sided die three times, what is the probability of the maximum outcome being 6?'

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#### Q.45

'Please calculate the probability of a red ball appearing on the first draw and the conditional probability of a white ball appearing on the second draw.'

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#### Q.46

'Throw a coin 9 times, find the probability of reaching point Q exactly on the 9th throw.'

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#### Q.47

'After playing 10 games of PRA and B, A won 7 times. Can we conclude that A is stronger than B based on this result? Use the concept of hypothesis testing and consider a significance level of 0.05. Note that there are no draws in the game.'

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#### Q.48

'In boxes A, B, and C, there are red, white, and black balls respectively. The quantities are as shown in the table on the right. Randomly select one box and draw one ball. Find the following probabilities.'

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#### Q.49

"Players A and B play the following game. They take out a ball from a bag containing 2 red balls and 1 white ball, check the color, and then return it. Depending on the color of the ball taken out, A gets 1 point if it's red, and B gets 2 points if it's white."

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#### Q.50

'Calculate the probability of event A occurring 8 or more times in a single trial.'

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#### Q.51

'Calculate the probability of getting heads when tossing 4 coins once, and calculate the expected value of X.'

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#### Q.52

'What is the probability of drawing at least 1 winning lottery ticket when drawing 2 tickets simultaneously from a lottery containing 3 winning tickets out of 10?'

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#### Q.53

'Find the probability of the product of the dice resulting in an odd number or a multiple of 12.'

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#### Q.55

'There are 20 cards, each with an integer from 1 to 20 written on it. (1) When two cards are drawn at the same time, what is the probability that the sum of the integers on the two cards is a multiple of 3? (2) When 17 cards are drawn at the same time, what is the probability that the sum of the integers on the 17 cards is a multiple of 3?'

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#### Q.56

'There are 20 lottery tickets in total, including 3 winning tickets. When drawing without replacement, find the following probabilities:\n(1) When A and B draw 1 ticket each in order, the probability of A losing and B winning\n(2) When A, B, and C draw 1 ticket each in order, the probability that only C wins'

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#### Q.57

'A and B are playing a game, where the first one to win 3 matches will be declared the winner. The probability of A winning a match is 1/3. Answer the following question. Assuming there are no draws. (2) Calculate the probability of A winning in the 4th match.'

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#### Q.58

'A and B are playing a game where the first to win 3 times is the champion. The probability of A winning in a single game is 1/3. Answer the following questions. Assume there are no ties. (3) Determine the probability of A winning.'

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#### Q.59

'Two players A and B play a game repeatedly. In each game, the probability of A winning against B is 2/3, and the probability of B winning against A is 1/3.\n1. If the first to win 3 times is declared the winner, then find the probability of A winning.\n2. When one player wins 2 more times than the other, the one with more wins is declared the winner. Find the probability of A winning by the end of the fourth game.'

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#### Q.60

'Choose a random four-digit integer between 1000 and 9999, and find the probability of having at least two identical digits.'

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#### Q.61

'In a P.O. box, there are 9 red number cards numbered 1 to 9 and 6 white number cards numbered 1 to 6. When drawing one card from this box, let the event of drawing an even number card be A, and the event of drawing a red card be B. In this case, calculate the following probabilities: \n(1) P(A ∩ B)\n(2) P_B(A)\nDefine the sample space as U. The number of cards in the box is as follows according to the table on the right:'

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#### Q.62

'When playing rock-paper-scissors with 3 people, calculate the probability of not ending in a tie.'

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#### Q.63

'Out of 15 light bulbs, 3 are defective. When 3 light bulbs are simultaneously drawn from these, find the probability of at least 1 defective bulb being included.'

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#### Q.64

'Explain how to determine the winning probability of each player in a competitive game.'

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#### Q.65

"Company X conducted a survey to determine whether their own product, pencil A, or another company Y's pencil B is easier to write with. Initially, two-thirds of all respondents said that A is easier to write with. Later, after company Y improved pencil B and conducted another survey, 14 out of 30 people said that A is easier to write with. Can we conclude that the ease of writing with A has decreased compared to B? Using the concept of hypothesis testing, consider each of the following cases. Use the example of the dice roll experiment mentioned above."

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#### Q.66

'Points A, B, C, D, E, F are arranged clockwise on the circumference. Throw a die, and if the outcome is 1 or 2, the moving point P advances two adjacent points clockwise, and if the outcome is 3, 4, 5, or 6, it advances one adjacent point counterclockwise. Starting from point A and throwing the die 5 times to move, find the probability of being at point B.'

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#### Q.67

'Please explain the probability addition rule when events A and B never occur simultaneously.'

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#### Q.68

'When three dice are rolled simultaneously, what is the probability that the sum of the numbers is 5?'

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#### Q.69

'Basic Example 57 Probability of Cause\nThere are machines A and B for manufacturing components, with defect rates of 3% for A and 5% for B. Suppose parts from A and B are mixed in a ratio of 7:3 and one part is selected from the mixture, with the event of it being defective denoted as E. In this case, calculate the following probabilities.\n(1) Probability P(E)\n(2) Probability that event E occurred due to machine A'

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#### Q.70

'When rolling three dice simultaneously, find the probability that the sum of the numbers is 5.'

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#### Q.71

'There are 3 red balls and 2 white balls in box A, totaling 5 balls, and there are 3 red balls and 4 white balls in box B, totaling 7 balls. When 2 balls are taken from A and B respectively, the probability that all 4 balls taken out are red balls is A, and the probability that 2 red balls and 2 white balls are taken out of the 4 balls is B.'

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#### Q.72

'When playing rock-paper-scissors with 3 people once, find the probability of a tie.'

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#### Q.73

'There are 2789 cards, each with the letters D, A, I, G, A, K, U written on them. After shuffling these 9 cards well, they are arranged in a single row. What is the probability of these cards being arranged from left to right in the order D, G, K, U when only the cards with D, G, K, U are seen? This probability is denoted by A. Also, the probability of three I cards being arranged consecutively is denoted by B.'

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#### Q.74

'(i) There are 2 cases in which the sum of the numbers on 3 cards is odd:'

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#### Q.76

'Throw 3 dice of sizes large, medium, and small at the same time, resulting in scores a, b, c respectively. Answer the following questions:\n(1) Find the probability that 1/a+1/b≥1.\n(2) Find the probability that 1/a+1/b≥1/c.'

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#### Q.77

'There are three examinees A, B, C. If the probability of each person passing their respective desired schools is 4/5, 3/4, and 2/3, respectively, find the following probabilities:'

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#### Q.78

'When two people play rock-paper-scissors once, what is the probability of determining the outcome?'

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#### Q.79

'When two dice are rolled simultaneously, let the smaller of the two numbers (or the number itself if they are equal) be denoted as X, and the larger one (or the number itself if they are equal) be denoted as Y. Given a constant a as an integer from 1 to 6, find the probabilities as follows: (1) X>a (2) X≤a (3) X=a (4) Y=a'

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#### Q.80

'In a certain experiment, when each elementary event is equally likely to occur, these elementary events are said to be equally certain. In such an experiment, if the total number of possible outcomes is N and the number of times event A occurs is a, find the probability of event A occurring, P(A).'

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#### Q.81

'There are 20 cards with integers from 1 to 20 written on them.'

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#### Q.82

'Basic Permutation Problem 44 Probability of Independent Trials\n(1) When throwing a die and flipping a coin simultaneously, find the probability of getting a number 4 or less on the die and the coin showing heads.\n(2) In bag A, there are 6 white balls and 4 black balls, and in bag B, there are 8 white balls and 2 black balls. When picking 3 balls from bag A and 2 balls from bag B, find the probability that all of them are white balls.\nPage 329 Basic Information 11\nC. HART \\& SOLUTION'

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#### Q.83

'When rolling two dice simultaneously, what is the probability that the minimum value is 3 or the maximum value is 4? There are a total of 36 possible outcomes.'

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#### Q.84

'In a bag, there are 3 white balls and 6 black balls. When 4 balls are simultaneously drawn from the bag, calculate the probabilities of the following events:\n(1) Drawing 1 white ball and 3 black balls.\n(2) Drawing 4 balls of the same color.'

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#### Q.86

'Sample Problem\nIn a general case, when drawing one card from 100 cards numbered from 1 to 100, find the probability that the number is a multiple of 3 or 4.'

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#### Q.87

'In a certain company, a pen B that improves the already existing pen A was developed. To evaluate the ease of writing, a survey was conducted on 20 randomly selected people to determine which one, A or B, is easier to write with. The result showed that 15 people chose B. Can it be concluded from this survey result that consumers appreciate B as easier to write with? Use a probability criteria of 0.05 and consider the results of the following coin toss experiment. Experiment: Toss a fair coin. Then, toss the coin 20 times in one set and record the number of times heads appear in each set. After repeating this experiment 200 times, the results are as follows.'

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#### Q.88

'Find the probability of getting 4 balls of the same color.'

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#### Q.89

'From a deck of 52 cards without the joker, A and B draw one card each in sequence. Calculate the following probabilities without replacement:\n(1) Both A and B draw a heart card\n(2) Only B draws a heart card'

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#### Q.91

'Explain probability and its basic properties, and calculate the probabilities for the following cases.'

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#### Q.92

'When arranging the 5 letters of DREAM in a row at random, find the probabilities for the following cases:\n(1) The right end is E.\n(2) A and D are adjacent.'

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#### Q.93

'When rolling two dice at the same time, find the probability that the sum of the numbers on the dice is 4.'

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#### Q.94

'From a bag containing 5 red beads and 4 white beads, what is the probability of drawing 4 beads of two different colors simultaneously?'

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#### Q.95

'When rolling 3 dice at the same time, calculate the following probabilities: (1) Probability of getting at least one odd number (2) Probability that the sum of the three numbers is not 4'

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#### Q.96

'When rolling 3 dice simultaneously, find the following probabilities:\n(1) Probability of getting at least one odd number\n(2) Probability that the sum of the three numbers is not 4'

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#### Q.97

'When one coin is tossed 6 times, find the following probabilities:\n(1) Probability of getting 4 or more heads\n(2) Probability of getting at least 1 head'

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#### Q.98

'A bag contains 6 red balls and 4 white balls. Find the probability that both a red ball and a white ball are drawn simultaneously when 3 balls are drawn from the bag.'

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#### Q.99

'Hypothesis testing is the process of determining whether a hypothesis about a population is statistically correct using data from obtained samples.'

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#### Q.00

'When rolling three dice simultaneously, find the following probabilities:\n(1) Probability that all outcomes are 3 or higher\n(2) Probability that the smallest outcome is 3'

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#### Q.01

'Out of 10 lottery tickets, 2 contain a winning prize. If you draw one ticket at a time and put it back, after drawing 4 times, what is the probability that the number of winning and losing tickets will be equal?'

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#### Q.02

'Investigate the percentage of responses by age group in a multiple-choice survey.'

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#### Q.03

'In a bag, there are 5 red balls and 4 black balls. When 3 balls are simultaneously drawn from this bag, find the following probabilities: (1) The probability that all 3 balls are of the same color. (2) The probability that only 2 balls are of the same color.'

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#### Q.04

'When a coin is tossed 3 times, what is the probability of getting at least one head?'

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#### Q.05

"Find the event of 'rolling an even number or a prime number' in the previous problem."

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#### Q.06

'Calculate the following probabilities:\n(1) Probability of getting exactly one head when flipping a coin three times\n(2) Probability of getting at least one head when flipping a coin three times\n(3) Probability of getting two or more consecutive heads when flipping a coin four times\n(4) Probability of not getting two consecutive heads when flipping a coin five times'

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#### Q.07

'Throw a die three times. Find the probability that the sum of the outcomes is 6.'

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#### Q.09

'When rolling a six-sided die four times, find the following probabilities: (1) Probability of the minimum value being 1 (2) Probability of the minimum value being 1 and the maximum value being 6'

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#### Q.10

'Obtain the probability of getting at least 3 white balls when picking a ball from a bag containing 3 white balls and 6 red balls, looking at the color, and then returning it to the bag for 4 consecutive trials.'

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#### Q.11

'In a company, a modified version of the pen A, called pen B, has been developed. In order to evaluate the ease of writing, a survey was conducted with 20 randomly selected individuals to determine which is easier to write with, A or B. The results showed that 12 people preferred B. Can it be concluded from this survey result that consumers perceive B to be easier to write with? The standard probability is set to 0.05 and considerations are made based on the results of the following coin flipping experiment. Flipping a fair coin and repeating the experiment 20 times in one set, recording the number of times heads appear in one set. After repeating this experiment for 200 sets, the results are as follows:'

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#### Q.12

'When rolling a die 4 times, find the following probabilities: (1) Probability of getting a minimum value of 1 (2) Probability of getting a minimum value of 1 and a maximum value of 6'

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#### Q.13

'There are 9 cards with numbers from 1 to 9 on them. When 3 cards are drawn simultaneously, find the probability that the sum of the numbers on the 3 cards is odd.'

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#### Q.14

'When tossing 3 coins simultaneously, what is the probability that all 3 coins will land heads up?'

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#### Q.15

'When throwing two dice of size 2 simultaneously, what is the probability that the product of the two dice results in a multiple of 10?'

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#### Q.16

'When drawing one ball from a bag containing one red, one blue, one yellow, and one white ball each, find the probability of drawing the red ball.'

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#### Q.19

'Find the probability of getting exactly 3 heads when tossing a coin 5 times.'

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#### Q.20

'From a bag containing 4 red balls and 3 white balls, calculate the following probabilities when drawing 2 balls at the same time: (1) the probability of both balls being red (2) the probability of drawing balls of different colors'

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#### Q.21

'Basic Example 38,000 describes a scenario in which 1 ticket is drawn from 3 boxes A, B, C, each containing winning tickets with probabilities of 1/4, 2/3, 1/2 respectively. When drawing 1 ticket from each box, calculate the following probabilities:'

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#### Q.22

'In a lottery of 12 tickets with 2 winning tickets, individuals A, B, C draw one ticket each in order. In the case of 44^3, calculate the probability that only A and C win when: (1) tickets are replaced after drawing (2) tickets are not replaced. The trials where A, B, C draw tickets are independent. A, B, C each draw one ticket from the 12 tickets including 2 winning tickets. Since A and C win while B loses, the probability is calculated as (2/12) * (10/12) * (2/12) = 5/216. Consider the probability of independent events with replacement.'

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#### Q.23

'In this set of 8 questions where circles represent correct answers and crosses represent incorrect ones, what is the probability of getting exactly 2 correct answers by randomly marking circles and crosses?'

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#### Q.24

'Three people A, B, and C play rock-paper-scissors once. Find the probability of A and B winning.'

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#### Q.25

'When throwing a die and a coin simultaneously, find the probability of the die showing an odd number and the coin showing tails.'

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#### Q.26

'A bag contains 6 red balls numbered from 1 to 6 and 5 blue balls numbered from 1 to 5. When one ball is drawn from the bag, find the probability that the number on the ball is odd or it is a blue ball.'

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#### Q.27

"Consider the following experiment: 'Simultaneously roll two dice'. Calculate the probability of 'at least one die shows a 6'."

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#### Q.28

'There is a bag containing 6 red balls numbered from 1 to 6 and 5 blue balls numbered from 1 to 5. Find the probability of picking a ball with an odd number or a blue ball when drawing one ball from the bag.'

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#### Q.29

'Calculate the probability of getting at least one 6 when rolling a dice.'

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#### Q.30

'When a coin is flipped three times, what is the probability of getting heads only once?'

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#### Q.31

'From a bag containing 5 red marbles and 4 white marbles, what is the probability of drawing 4 marbles at the same time with two different colors?'

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#### Q.32

'When throwing 3 dice simultaneously, calculate the following probabilities:\n(1) The probability that all dice show 3 or more\n(2) The probability that the smallest number rolled is 3\n[Source: Shiga University]'

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#### Q.33

'When playing rock-paper-scissors with 3 people, determine the probability of a tie.'

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#### Q.34

'When rolling two dice simultaneously, find the following probabilities:\n(1) Probability of not getting the same number\n(2) Probability of getting at least one even number'

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#### Q.36

'Let n be a natural number, and there are a total of (2n+1) cards with numbers 0, 1, 2, ⋯, 2n written on them, each number appearing only once. Randomly pick one card from them, let the number written on it be X. Find the probability that Y=k for all k=0, 1, 2, ⋯, 2n when Y is defined according to the following steps. Also, find the variance of Y. (a) If X is odd, then Y=X. (b) If X is even (including 0), then put the card back, randomly pick one card again from all the cards, and let the number on it be Y.'

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#### Q.37

'A survey was conducted on 600 sixth-grade students randomly selected from a certain local elementary school, of which 262 were found to have cavities. It is claimed that the proportion of sixth graders with cavities nationwide is 40%. Test whether the proportion of sixth graders with cavities in this local elementary school is higher than the national average at the following significance levels.'

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#### Q.38

'There are 9 cards numbered 1 to 9. From these cards, 4 cards are drawn continuously without replacement, denoted as a, b, c, d in order. (1) Find the probability that the product of a b c d is even. (2) Let the thousands place of the card be a, the hundreds place be b, the tens place be c, and the ones place be d, find the expected value of the 4-digit number N obtained. [Akita]'

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#### Q.39

'It cannot be determined that the probability of rolling a 671 on a die is 1/6.'

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#### Q.40

'What points should be checked when purchasing digital educational materials?'

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#### Q.41

"A survey of 600 sixth graders randomly selected from a certain region's elementary school revealed that 262 had cavities. It is claimed that the proportion of sixth graders with cavities in the entire country is 40%. Can it be said that the proportion of sixth graders with cavities in this region's elementary school is higher than the national level? Test at the following significance levels."

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#### Q.42

'When the probability density function of a random variable X is given by f(x)=1-\x0crac{1}{2} x (0 ≤ x ≤ 2), find the specified probabilities.'

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#### Q.43

'The sample mean is X̄=6.5, the population standard deviation is σ, and the sample size is n=250. Therefore, the 95% confidence interval for the population mean m is [6.5-1.96⋅σ/√250, 6.5+1.96⋅σ/√250]'

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#### Q.44

'In a certain experiment, a variable whose value is determined by the outcome of the experiment and has probabilities defined for each value is called a random variable. Generally, when a random variable X can take values x1, x2, ..., xn, and the corresponding probabilities are p1, p2, ..., pn, the following holds true.'

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#### Q.45

'On n cards, numbers 1, 2, 3, ..., n are written one by one. When two cards are randomly drawn from these cards, let the smaller number be X and the larger number be Y. It is assumed that n ≥ 2. (1) Find the probability that X = k, where k = 1, 2, 3, ..., n. (2) Find the expected value of X. (3) Find the variance of Y.'

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#### Q.46

'Basic Problem 75 Law of Large Numbers\nFind the probability that the sample mean X̄, extracted from a population with a population mean of 0 and a population standard deviation of 1, is between -0.1 and 0.1 for each case of n=100, 400, 900.'

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#### Q.47

'In a certain region A, the heights of 400 15-year-old boys were measured, resulting in an average of 168.4 cm and a standard deviation of 5.7 cm. Find the 95% confidence interval for the average height m cm of 15-year-old boys in A.'

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#### Q.48

'In region Y, the support rate of party B was 1/3. Party B proposed a policy and suspected a change in the support rate. Therefore, a survey was conducted on 30 people, with 15 people supporting party B. Based on this result, can it be concluded that the support rate of party B has increased? Analyze each case using the concept of hypothesis testing. Suppose that an experiment was conducted 200 times by throwing a fair 30-sided dice, and the results of the number of occurrences of 1 to 4 are as shown in the table:'

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#### Q.49

"The scatter plot on the right is a scatter plot of test scores out of 100 for 187 kanji and English words in a class of 30 students. (1) Based on this scatter plot, investigate whether there is a correlation between the scores of kanji and English words. If there is a correlation, state whether it is positive or negative. (2) Based on this scatter plot, create a frequency distribution table for English words. The class is set as 'full'."

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#### Q.50

'It can be judged that this die is more likely to roll a 1.'

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#### Q.52

'A and B played a game 9 times. A won 7 times. Can we conclude that A is stronger than B based on this result? Using the concept of hypothesis testing, discuss with a significance level of 0.05. Assuming there are no ties in the game.'

#### A. ...

#### Q.53

'Calculating mean and variance through data integration'

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#### Q.54

'After playing 10 games, A won 7 times. Can we conclude that A is stronger than B based on this result? Using the concept of hypothesis testing with a significance level of 0.05. Assume there are no ties in the games.'

#### A. ...

#### Q.55

"A company created a mascot and conducted a survey of 20 people, of which 13 responded with 'the company's image has improved'. Can we conclude that the company's image has indeed improved? Using the concept of hypothesis testing, consider a significance level of 0.05. However, it is known that an experiment of flipping a fair coin 20 times was conducted 200 times, and the results are as shown in the table below. Please use these results for your analysis."

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#### Q.56

'Exercise 22\n(1) After the first operation, the marked face will always come to the side face, so the probability that the marked face will consecutively come to the side face after the next operation is\n\\\\n\\\\frac{2}{4}=\\\\frac{1}{2}\n\\\'

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#### Q.57

"Of these four stages, the main determinant of success or failure is likely the guidelines in the second stage. The specific approach can be seen in the chart in the text, but first let's address some general considerations."

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#### Q.58

'To calculate the probability of the leading digit being k, calculate the range width Lk that includes n.'

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#### Q.59

'Prove that after rolling a die 2n times, point P is at either A or C (proposition 1).'

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#### Q.60

'When throwing 12 dice simultaneously, what is the probability that the sum is a prime number?'

#### A. ...

#### Q.61

'Example 23 | Combinations and Probabilities\nThere are 4 cards each of red, blue, and yellow, with numbers 1 to 4 written on each card. When 3 cards are randomly drawn from these 12 cards, find the probabilities of the following events:\n(1) All cards are of the same color.\n(2) All numbers are different.\n(3) All colors and numbers are different.\n[Saitama Medical University]'

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#### Q.65

'Calculate the probability of getting 3 heads when flipping a coin 6 times.'

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#### Q.66

'When rolling a die once, the probability of getting a prime number is \\\frac{3}{6}\, and the probability of getting a non-prime number is also \\\frac{3}{6}\. \\({}_{5}\\mathrm{C}_{4}\\left(\\frac{3}{6}\\right)^{4}\\left(\\frac{3}{6}\\right)^{1}=5\\times\\left(\\frac{1}{2}\\right)^{5}=\\frac{5}{32}\\)\n(i) The event of getting at least 4 prime numbers is when getting 4 or 5 prime numbers, so the probability is \n\\(\\frac{5}{32}+\\left(\\frac{3}{6}\\right)^{5}=\\frac{5}{32}+\\frac{1}{32}=\\frac{3}{16} \\)'

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#### Q.69

'From the event that all drawn cards are 7 or less, except the event that all are 6 or less, find the probability of drawing a maximum value of 7 when drawing one card.'

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#### Q.70

'Probability of a point moving around the perimeter of 31 figures'

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#### Q.71

'Example 24 | Rock-Paper-Scissors Probability\nWhen 4 people play rock-paper-scissors once, find the following probabilities:\n(1) Probability of only one person winning\n(2) Probability of 2 people winning\n(3) Probability of a tie'

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#### Q.75

'The probability of scoring a goal with one shot is \ \\frac{2}{3} \'

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#### Q.76

'From a bag containing 3 red balls and 3 blue balls, two balls are drawn, their colors checked, and then returned to the bag. This process is repeated 2 times. Find the probability that out of the 4 balls drawn after 2 trials, 2 are red and 2 are blue.'

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#### Q.77

'A die was rolled 8 times, and an even number came up 7 times. Can it be concluded from this result that the die is biased towards even numbers? Discuss using the principles of hypothesis testing with a significance level of 0.05.'

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#### Q.79

'(2) In one trial, moving the stone counterclockwise is represented by +, moving it clockwise is represented by -, and not moving is represented by 0. After 4 trials, there are 5 combinations where the stone is at C:\n[1] (+,+,0,0)\n[2] (+,+,+,-)\n[3] (-,0,0,0)\n[4] (-,-,+ ,0)\n[5] (-,-,-,-)\nProbability of one iteration'

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#### Q.82

'What are the characteristics of studying mathematics, and how does it compare to other subjects?'

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#### Q.83

'When a bag containing 5 red gems, 4 white gems, and 3 blue gems is selected, what is the probability of drawing a red gem?'

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#### Q.84

'When throwing 3 coins at the same time, determine the probability of getting 1 head and 2 tails.'

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#### Q.86

'The event of the sum of numbers on dice being 6 is the union event of events A, C, and D, which are mutually exclusive.'

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#### Q.88

'When rolling two dice, what is the probability that the larger die is an odd number and the smaller die is 5 or higher?'

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#### Q.89

'Find the probability of a single winner being determined in the nth round.'

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#### Q.90

'The probability of A winning in each match is 1/3, and the probability of B winning is 2/3.\nFor A to win, there are three cases:\n12 trials are independent and probabilities can be multiplied.\nAddition rule (adding probabilities)\nExcept for prime numbers, once ✔️\n${ }_{5} \\mathrm{C}_{4}\\left(\\frac{3}{6}\\right)^{4}\\left(\\frac{3}{6}\\right)^{1}$\n4 primes out of 5 occurrences\n4 primes out of 5 occurrences\nUsing the addition rule.\n2 out of 3 are determined.\nChange perspective for difficult probabilities (complementary events).\n${}_{6} \\mathrm{C}_{3} \\times{ }_{3} \\mathrm{C}_{1} \\times{ }_{2} \\mathrm{C}_{2}$ works too.\nMultiply the probabilities for the case of 60 possibilities where X occurs 3 times, Y occurs 1 time, and Z occurs 2 times.\nThe probability of B winning is 1-1/3'

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#### Q.91

'From a deck of 52 cards excluding the joker, when picking 1 card, which two events are mutually exclusive among the events A: drawing an Ace, B: drawing a Heart, C: drawing a face card?'

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#### Q.92

'There are 27 cards numbered from 1 to 9, with 3 cards for each number. After shuffling well, when drawing 2 cards, find the following probabilities:\n(1) Probability of getting two cards with the same number\n(2) Probability of getting two cards with the same number or the sum of the two numbers not exceeding 5.'

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#### Q.96

'After playing a game between A and B 10 times, A won 8 times. Can we conclude from this result that A is stronger than B? Use the concept of hypothesis testing with a significance level of 0.05 for analysis. Assume there are no ties in the game.'

#### A. ...

#### Q.97

'Throw a die 3 times, let X be the product of all outcomes. Find the probability that X is greater than 2.'

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#### Q.00

"Sometimes the likelihood of an event occurring is expressed as a number. This number is referred to as 'probability'. Based on the definition of probability, this chapter discusses the theorems and properties of probability (including the addition theorem of probabilities, multiplication theorem, probability of independent events, etc.), and using concepts learned in previous chapters such as permutations and combinations, learns how to calculate probabilities. Furthermore, it also explores the concept of the average value of numerical outcomes of experiments—also known as expected value."

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#### Q.01

"When the three archers A, B, and C shoot one arrow at the target, the probabilities of hitting the target are 1/2, 1/3, and 1/4 respectively. Since Γ×1/2 + 1/3 + 1/4 = 13/12 > 1, the idea that 'at least one person hits the target' is correct."

#### A. ...

#### Q.03

'Example 32 Probability of Maximum and Minimum Values\nWhen a die is rolled 4 times, find the following probabilities.\n(1) The probability that the minimum value is 3.\n(2) The probability that the minimum value is 1 and the maximum value is 6.'

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#### Q.05

'Find the probability of A winning 4 matches in a row and securing the championship after the second match.'

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#### Q.06

'Events [1] to [3] are mutually exclusive, therefore calculate the required probability.'

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#### Q.08

'Find the total number of ways and probability of drawing 3 numbers that sum up to 0.'

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#### Q.09

'There are 36 possible outcomes when rolling two dice (ways).'

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#### Q.13

Is it the 'common range' or the 'combined range'? In the answer to example 44 on the previous page, there are instances of asking for the 'common range' and instances of asking for the 'combined range.' Let's take a closer look at the difference based on the answer to example 44.