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Two events $A$ and $B$ have probabilities respectively of $p(A)$ and $p(B)$.

What is the probability that exactly one event will occur if

  1. $A$ and $B$ are mutually exclusive
  2. $A$ and $B$ are not mutually exclusive, but dependent
  3. $A$ and $B$ are not mutually exclusive, but independent

Attempt:

  1. $P(A)+P(B)$
  2. $P(A)P(B^c)+P(B)P(A^c)$
  3. $P(A\cap B^c)+P(A^c\cap B)$
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  • $\begingroup$ You mixed 2 and 3, and the independent case could be simplified a bit $\endgroup$ Jul 11, 2017 at 4:46
  • $\begingroup$ $P(A \cap B^c) + P(A^c \cap B)$ is true for all three, but in case (2) it can't be simplified. $\endgroup$ Jul 11, 2017 at 4:51

1 Answer 1

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  1. $ P(A\cap B^c) + P(B\cap A^c) = P(A) + P(B) $

  2. $P(A\cap B^c) + P(B\cap A^c) = P(A\cup B) - P(A\cap B) $

  3. $P(A\cap B^c) + P(B\cap A^c) = P(A\cup B) - P(A\cap B) = P(A) + P(B) - 2P(A)P(B)$

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