Applied-Probability-and-Statistics Leak Questions

For two events, the general addition rule is P(A or B) = P(A) + P(B) − P(A and B). Because A and B are independent, their joint probability is P(A and B) = P(A)P(B). Here, P(A) = 0.3 and P(B) = 0.2, so P(A and B) = 0.3 × 0.2 = 0.06. Then P(A or B) = 0.3 + 0.2 − 0.06 = 0.44. The subtraction is necessary because the overlap, where both A and B occur, is counted twice when adding P(A) and P(B). Option A, 0.5, incorrectly adds the two probabilities without correcting for overlap. Option B gives the probability of both events, not either event. Option D is unrelated to the addition rule. Study Guide references/topics: independent events, addition rule, multiplication rule, union of events.

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When two standard six-sided dice are rolled, there are 6 × 6 = 36 equally likely ordered outcomes. To get a sum of 2, the only possible ordered pair is (1, 1). No other pair of die faces can produce a total of 2 because 1 is the smallest value on each die. Therefore, there is 1 favorable outcome out of 36 total outcomes, giving a probability of 1/36. Option B, 1/6, is the probability of one specific result on a single die, not a two-dice sum. Option C, 2/36, would require two favorable ordered pairs, but only one exists. Option D, 1/12, is also too large because it corresponds to 3 outcomes out of 36. The key is to count ordered outcomes in the full two-dice sample space. Study Guide references/topics: dice probability, sample space, favorable outcomes, theoretical probability.

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“At least one six” means one six or two sixes across the two rolls. The most efficient method is to use the complement rule. The complement of at least one six is no six on either roll. On one die roll, the probability of not rolling a six is 5/6. Since the two die rolls are independent, the probability of no six on both rolls is (5/6)(5/6) = 25/36. Therefore, the probability of at least one six is 1 − 25/36 = 11/36. This includes all outcomes with a six on the first roll, a six on the second roll, or sixes on both rolls. Option B gives the probability of a six on one single roll only. Option C gives the probability of rolling two sixes. Option D gives the complement, no sixes. Study Guide references/topics: complement rule, independent events, dice probability, sample spaces.

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A z-test for a population mean is used when the population standard deviation σ is known and the sampling distribution of the test statistic can be treated as normal. The test statistic has the form z = (sample statistic − hypothesized parameter) divided by the standard error. Knowing σ allows the standard error to be computed using σ/√n rather than estimating it with the sample standard deviation. Option B describes the common setting for a t-test, not a z-test. Option C is not sufficient by itself; small samples generally require stronger normality assumptions and often favor t-procedures when σ is unknown. Option D is unrelated to the classic z-test for a mean, though z-tests can also be used for proportions under appropriate large-sample conditions. In the provided answer set, the defining condition is that the population standard deviation is known. Study Guide references/topics: z-test, population standard deviation, standard error, hypothesis testing.