XXXV OM - I - Problem 9

Three events satisfy the conditions:
a) their probabilities are equal,
b) any two of them are independent,
c) they do not occur simultaneously.
Determine the maximum value of the probability of each of these events.
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Suppose that $ A $, $ B $, $ C $ are these events, each with a probability of $ p $. Given the assumption of independence between any two of the considered events, the probability of each intersection $ A \cap B $, $ B \cap C $, $ C \cap A $ is $ p^2 $. Furthermore, $ A \cap B \cap C \ne 0 $. Therefore,

Since of course $ P (A \cup B \cup C) \geq P (A \cap B) + P (B \cap C) + P (C \cap A) = 3p^2 $, it follows that $ 3p-3p^2 \geq 3p^2 $, which implies that $ p \leq \frac{1}{2} $.
The value $ p = \frac{1}{2} $ is assumed, for example, by the following events. When flipping a coin twice, we take $ A $ to be the event that a tail appears on the first flip, $ B $ - that a head appears on the second flip, $ C $ - that the same result appears on both flips. Clearly, $ P(A) = P(B) = P(C) = \frac{1}{2} $, the events $ A $, $ B $, $ C $ are pairwise independent and cannot occur simultaneously.
Therefore, the maximum value of the considered probability is $ \frac{1}{2} $.