For the three events A, B and C, P (exactly o | vedclass

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Question

(a) We know that $P$ (exactly one of $A$ or $B$ occurs)
= $P(A) + P(B) - 2P(A \cap B)$
Therefore, $P(A) + P(B) - 2P(A \cap B) = p$&hellip;..$(i)$
S

Vedclass · Accepted Answer

$\frac{{3p + 2{p^2}}}{2}$

Vedclass · Answer

(a) We know that $P$ (exactly one of $A$ or $B$ occurs)
= $P(A) + P(B) - 2P(A \cap B)$
Therefore, $P(A) + P(B) - 2P(A \cap B) = p$&hellip;..$(i)$
Similarly, $P(B) + P(C) - 2P(B \cap C) = p$&hellip;..$(ii)$
and $P(C) + P(A) - 2P(C \cap A) = p$&hellip;..$(iii)$
Adding $(i),$ $(ii)$ and $(iii),$ we get
$P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(C \cap A) = \frac{{3p}}{2}$
.....$(iv)$
We are also given that $P(A \cap B \cap C) = {p^2}$ .....$(v)$
Now, $P$ (at least one of $A,\,\,B$ and $C)$
$ = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C)$
$ - P(C \cap A) + P(A \cap B \cap C)$
$ = \frac{{3p}}{2} + {p^2}$, [By $(iv)$ and $(v)$].

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