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PROBABILITY — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSPROBABILITY3 Marks
Question
If A, B, C are three events associated with a random experiment, prove that
$P(A\cup B\cup C) = P(A) + P(B) +P(C) - P(A\cap B) - P(A\cap C) - P(B\cap C) + P(A\cap B\cap C)$
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Answer

Consider E = $B \cup C$.
So P ($A \cup B \cup C$) = P ($A ∪ E$)
= P(A) + P(E) - $P(A\cap E)$ ..................(1)
Now P(E) = $P(B\cup C)$ .......................(2)
Also $(A\cap E) = A\cap (B\cup C) = (A\cap B)\cup (A\cap C)$ [using distribution property of intersection of sets over the union].
Writing them as probability,
$P(A\cap E) = P(A\cap B) (A\cup C) - P[(A\cap B)\cap (A\cap C)]$
$P(A\cap B) + P(A\cap C) - P[A\cap B\cap C]$..........................(3)
using (2) and (3) in (1), we get
$P[A\cup B\cup C] = P(A) + P(B) + P(C) - P(B\cap C)- P(A\cap B) - P(A\cap C) + P(A\cap B\cap C)$
Hence Proved.

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