First slide

Theorems of probability

Question

$Let A, B, C be three events. If the probability of occurring exactly one event out of A and B is 1 - a, out of B$
$and C is 1 - 2 a, out of C and A is 1 - a and that of occurring three events simultaneously is a^{2}, then the probabil ity$ $that at least one out of A, B, C will occur is$

Moderate

A

½
B

Greater than ½
C

Less than ½
D

Greater than ¾

Solution

$P (exactly one event out of A and B occurs) = P [A \cap B^{'}) \cup (A^{'} \cap B)]$
$= P (A \cup B) - P (A \cap B) = P (A) + P (B) - 2 P (A \cap B)$
$∴ P (A) + P (B) - 2 P (A \cap B) = 1 - a - - - - (1)$
$Similarly, P (B) + P (C) - 2 P (B \cap C) = 1 - 2 a - - - - (2)$
$\begin{array}{l} P (C) + P (A) - 2 P (C \cap A) = 1 - a - - - - (3) \\ P (A \cap B \cap C) = a^{2} - - - (4) \end{array}$
$n o w, \begin{aligned} P (A \cup B \cup C) \end{aligned}$
$= P (A) + P (B) + P (C) - P (A \cap B) - P (B \cap C) - P (C \cap A) + P (A \cap B \cap C)$
$= \frac{1}{2} [P (A) + P (B) - 2 P (B \cap C) + P (B) + P (C) - 2 P (B \cap C) + P (C) + P (A) - 2 P (C \cap A)] + P (A \cap B \cap C)$
$\begin{array}{r} = \frac{1}{2} [1 - a + 1 - 2 a + 1 - a] + a^{2} \\ [using (1), (2), (3) and (4)] \\ = \frac{3}{2} - 2 a + a^{2} = \frac{1}{2} + (a - 1)^{2} > \frac{1}{2} \end{array}$

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