P(A) is the probability that something occurred. If it occurred again would the probability of both occurring (intersection) be P(A)^2 or would it be 1 - P(A')^2? What are the conditions that would distinguish one from the other?
2026-04-11 03:27:50.1775878070
Given P(A) intersecting itself, when does the resulting probability equal P(A)^2 versus 1 - P(A')^2?
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Suppose we have two independent trials, with an identical probability for an event happening in each particular trial. Let $\mathsf P(A)$ be that probability.
Then the probability that the event occurs in both trials will be $\mathsf P(A)^2$, by the independence.
Whereas $(1-\mathsf P(A))^2$ would be the probability that the event does not occur in both trials.
Also $2\,\mathsf P(A)\,(1-\mathsf P(A))$ would be the probability for the event occuring exactly one trial.
And $1-(1-\mathsf P(A))^2$ the probability for at least one occurance among the two trials.