If P(A) is the probability of event A occurring, what is the probability of the complement of A, denoted as P(A′)?

Question

BlackTom AI · Accepted Answer

We start by restating the problem: Given P(A) as the probability of event A, determine P(A′), the probability of the complement of A.

Option 1: 'P(A) - 1' would yield a value that is typically negative (since 0 ≤ P(A) ≤ 1, P(A) - 1 ≤ 0). A probability cannot be negative, so this option is implausible as a general formula for a probability measure.

Option 2: 'P(A) + 1' would give a value greater than 1 for any nonzero P(A), and probabilities cannot exceed 1. This makes it an invalid expression for a probability in the standard framework.

Option 3: 'P(A) / 2' halves the probability of A. While this could be true in a specific context, it does not represent the probability of the complement, which depends on the probability of A in a precise way. There is no general identity that equates P(A′) to P(A) divided by 2.

Option 4: '1 - P(A)' aligns with the fundamental complement rule in probability: the probability that A does not occur is one minus the probability that A does occur, assuming the usual 0 ≤ P(A) ≤ 1 framework. This is the standard expression for the complement.

Putting these together, we see that only the fourth option matches the formal definition of a complement probability, while the first three violate basic probability constraints or the complement relationship. The reasoning hinges on the basic axiom that the probabilities of mutually exclusive complementary outcomes sum to 1, which yields P(A) + P(A′) = 1 and thus P(A′) = 1 − P(A).

If P(A) is the probability of event A occurring, what is the probability of the complement of A, denoted as P(A′)?多项选择题

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