Suppose that two fair dice are tossed. Let Ai be the event that a 6 shows on the ith die, i=1,2. Then P(A1∩A2) is:

Question

BlackTom AI · Accepted Answer

Question restatement: We toss two fair dice and define A1 as the event that the first die shows a 6, and A2 as the event that the second die shows a 6. We want P(A1 ∩ A2).

Option analysis:
- Option "1/12": To get 1/12, one would typically multiply probabilities that are not independent or miscount combinations. Here, the events A1 and A2 concern separate dice, and their joint occurrence is not obtained by simply adding probabilities of independent events in this way. This value overestimates the joint likelihood when dealing with two separate dice.
- Option "1/36": This matches the standard rule for independent events: P(A1 ∩ A2) = P(A1) × P(A2). Each die independently shows a 6 with probability 1/6. Thus, (1/6) × (1/6) = 1/36. Conceptually, the chance that both dice show 6s is the product of the individual chances, since the outcome of one die does not affect the other.
- Option "1/6": This could arise if one incorrectly considered only one die’s probability or erroneously treated the two events as mutually exclusive or dependent in a way that ignores the second die’s contribution. But P(A1 ∩ A2) cannot equal 1/6 because that would imply the second die always matches the first in a way that doubles the effect of a single-die probability, which is not how joint probabilities work for independent dice.
- Option "0": This would imply that it is impossible for both dice to show 6 simultaneously. Since each die has a nonzero probability of showing 6, and they are independent, both showing 6 has a positive probability, so 0 is incorrect.

Summary of reasoning: Because A1 and A2 refer to independent outcomes of two distinct dice, the probability that both show 6 is the product of their individual probabilities, (1/6) × (1/6) = 1/36. Other options arise from common misconceptions such as misapplying independence, adding probabilities, or assuming impossibility.

Suppose that two fair dice are tossed. Let Ai be the event that a 6 shows on the ith die, i=1,2. Then P(A1∩A2) is: 单项选择题

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