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## Statistics and Probability

### Fundamentals of Probability - Sum of Probabilities (Addition Rule)

#### Q.01

'Explain the probability addition theorem using permutation methods.'

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'In general, the following addition rule holds.\nAddition Rule\nLet two events A and B not occur simultaneously. If there are a ways for A to occur and b ways for B to occur, then\nThe total number of ways A or B can occur is a + b\nThe addition rule also holds for three or more events.'

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'Provide the formula that satisfies the following relationship.'

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

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'Topic: Sum and product of random variables, binomial distribution'

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'A mathematical problem is as follows:'

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'The complement of event A, \ar{A}, is the event where neither card is a heart, so P(A)=1-P(\ar{A})=1-\\frac{_{39} \\mathrm{C}_{2}}{_{52} \\mathrm{C}_{2}}=1-\\frac{19}{34}=\\frac{15}{34}. For event B, there are _{4} \\mathrm{C}_{2} ways to select two different suits, and each suit has 13 possible cards, so P(B)=\\frac{_{4} \\mathrm{C}_{2} \\times 13^{2}}{_{52} \\mathrm{C}_{2}}=\\frac{6 \\cdot 13^{2}}{26 \\cdot 51}=\\frac{26}{34}(=\\frac{13}{17}). Furthermore, event A \\cap B is when one card is a heart and the other card is a non-heart suit, so P(A \\cap B)=\\frac{_{13} \\mathrm{C}_{1} \\times_{39} \\mathrm{C}_{1}}{_{52} \\mathrm{C}_{2}}=\\frac{13 \\cdot 39}{26 \\cdot 51}=\\frac{13}{34}. Therefore, the required probability is P(A \\cup B)=P(A)+P(B)-P(A \\cap B)=\\frac{15}{34}+\\frac{26}{34}-\\frac{13}{34}=\\frac{14}{17}.'

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'Please explain with an example how the law of addition holds.'