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## Statistics and Probability

### Fundamentals of Probability - Probability Distributions (Discrete, Continuous)

#### Q.01

'In a certain country, it is said that the distribution of blood types among its citizens is O type 30%, A type 35%, B type 25%, and AB type 10%. Now, when randomly selecting 400 people, find the probability that the number of AB type individuals is between 37 and 49.'

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'Find the probability distribution, expected value E(X), variance V(X), and standard deviation σ(X) of the number of heads X when throwing two coins.'

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'Translate the given text into multiple languages.'

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'A random variable X can take any value in the interval [0,10], and its probability density function is given by f(x)=kx(10-x) (where k is a constant). In this case, k='

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'Distribution of sample means\nWhen a random sample of size n is taken from a population with a population mean m and population standard deviation σ, the sample mean X̅ approximately follows a normal distribution N(m, σ^2/n).'

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'When a random variable X follows a normal distribution N(m, σ^2), find P(|X-m| ≥ σ/4). Round the answer to the fourth decimal place.'

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'Prove that the number of times event A occurs, X, in n trials where the probability of event A occurring is p, follows a binomial distribution. Also, calculate the mean, variance, and standard deviation.'

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'When the proportion of elements in a population with a certain characteristic A (population proportion) is p, what distribution can the sample proportion R be considered to follow?'

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"Let's deepen our understanding of the binomial distribution."

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'For a binomial distribution B(n, p) with a random variable X, show the approximate normal distribution formula that X follows when n is large. Let q = 1 - p.'

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'When the random variable X follows the normal distribution N(m, σ^2), show the transformation formula to obtain the standard normal distribution Z.'

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'(1) When the random variable Z follows the standard normal distribution N(0,1), find the probability P(-1.98 ≤ Z ≤ -0.5).\n(2) When the random variable X follows the normal distribution N(30,4²), find the probability P(22 ≤ X ≤ 32).'

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'Using the normal distribution table, solve the following problem: Find the probability when u = 0.4 and z = 0.73.'

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'The defect rate of a certain product is approximately 7%. How many products need to be sampled to ensure that the width of the confidence interval for the defect rate, with a confidence level of 95%, is less than or equal to 4% and 2%, respectively?'

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'Please calculate the percentage (%) of deviation values y greater than 65. Given that the mean value of y is 50 and the standard deviation is 10, let z=(y-50)/10, then z follows the standard normal distribution N(0,1).'

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'Since the sample size n is 400, the 95% confidence interval for the population mean m is 51.0-1.96*(9.5 / sqrt(400)) <= m <= 51.0 + 1.96*(9.5 / sqrt(400)), therefore 50.069 <= m <= 51.931'

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'When two dice are rolled simultaneously, find the probability distribution of the random variable X. If the two dice have the same number, that number will be considered as X.'

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'When the probability density function f(x) of the random variable X is given by the following equation, find the specified probabilities.'

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'Prepare 3 cards with the number 1, 3 cards with the number 2, and 3 cards with the number 3, totaling 9 cards. When randomly selecting 3 cards from these, what is the probability that the sum of the numbers on the cards is a multiple of 3?'

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"Please demonstrate that events A and B become mutually exclusive when event B is changed to 'ball (3) appears'."

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"The event of 'at least 2 girls standing consecutively' is the complementary event to 'girls not being next to each other'. Girls not being next to each other occurs when 3 girls are standing between 10 boys. The total number of circular permutations of 10 boys is (10-1)!=9! (ways). For each of these cases, there are 10P3 ways for 3 girls to stand in 3 out of 10 positions. Therefore, the number of ways for girls not being next to each other is 9! × 10P3 (ways). Hence, the required probability is P(̅A) = 1 - P(A) = 1 - (9!×10P3/12!) = 1 - 10×9×8/12×11×10 = 5/11."

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"Let's assume there is a biased die with probabilities of rolling 1, 2, 3, 4, 5, 6 as 1/6, 1/6, 1/4, 1/4, 1/12, 1/12 respectively. Calculate the probability of getting a sum of 6 when rolling the die three times in a row. [Tokyo Denki University]"

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'Translate the given text into multiple languages.'

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"Suppose 4 people a, b, c, d from probability distribution A are divided into two groups {a, b} and {c, d}, and each person must choose one person from the other group with equal probability. Assume that each person's choice is independent. Let X be the number of pairs who choose each other. Find: (1) The distribution of the random variable X. (2) Find the expected value of X."

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'Basic Example 62 Binomial Distribution Mean and Variance\nFrom a bag containing 6 red balls and 4 white balls, one ball is drawn and then replaced. Repeat this process 6 times, let X be the number of times a red ball appears, calculate the expected value E(X), variance V(X), and standard deviation σ(X) of X.'

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'Assuming the grades follow a normal distribution with m = 62, σ = 20, what grade will a student who scored 85 receive?'

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'Item: Statistical Inference\nTopic: Probability Distribution\nQuestion Number: 6\nQuestion Content: Explain about probability distribution.'

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'Consider all students in a high school as the population, with a population proportion of students who never read books being 0.5 and a random sample size of 100. When the random variable X follows a binomial distribution B(100, 0.5), calculate the mean (expected value) and standard deviation of X. Also, assuming a population proportion of 0.5 for students who never read books, show the probability when X approximately follows a normal distribution.'

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'Roll two dice simultaneously, let X be the minimum of the two results, determine the probability distribution of X. Also, calculate P(X ≤ 3).'

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'The probability of rolling a 1 with one throw of a die is 1/6.'

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'Let X be a random variable following a binomial distribution with mean 6 and variance 2. Let Pk be the probability when X = k. Find the value of P4/P3.'

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'Find the value of the positive constant a when the probability density function of random variable X is given by f(x).'

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'Basic Example 68 Use of the Normal Distribution\nLet X be the height of male students in a certain high school, following a normal distribution with mean 170.9 cm and standard deviation 5.4 cm. Answer the following questions. Round to one decimal place.\n(1) What percentage of students have a height of 175 cm or higher?\n(2) What height is required to be taller than approximately 4% of students?'

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'If a new drug was used on 400 patients and 8 of them experienced side effects, can we say that the rate of side effects of this new drug is not 4% assuming the rate of side effects for the drug that has been used traditionally is 4%? Perform a hypothesis test with a significance level of 5%. Also, how about at a significance level of 1%? Assume that the 400 patients were randomly sampled.'

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'When the random variable X follows a normal distribution N(15,3^2), find the following probabilities:\n(1) P(X ≤ 18)\n(2) P(6 ≤ X ≤ 21)'

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'At university A, 64% of all students supported issue X. At another university B, out of 400 randomly selected students, 274 students supported X. Can it be said that there is a difference in the support rate for X between students at B and students at A? Test at a significance level of 5%.'

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'Explain how to find a random variable Z that follows a standard normal distribution.'

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'Assuming the distribution of exam scores follows a normal distribution with a mean of 58.4 and a standard deviation of 25. Please calculate the probability that the average score of 100 randomly selected individuals taking the test is at least 62 points.'