QJEEM17Ma0df7: JEE Main 2017 Mathematics - Class 12 JEEM17Ma0df7 Solution: For three events A, B and C, P(Exactly one of A or B occurs) =…

Given, P (A $\cap$ B $\cap$ C) = ${1 \over {16}}$

P (exactly one of A or B occurs)

= P(A) + P (B) – 2P (A $\cap$ B) = ${1 \over 4}$ .....(1)

P (Exactly one of B or C occurs)

= P(B) + P (C) – 2P (B $\cap$ C) = ${1 \over 4}$ .....(2)

P (Exactly one of C or A occurs)

= P(C) + P(A) – 2P (C $\cap$ A) = ${1 \over 4}$ .....(3)

Adding (1), (2) and (3),we get

2[ P(A) + P(B) + P (C) - P (A $\cap$ B)

- P (B $\cap$ C) - P (C $\cap$ A)] = ${3 \over 4}$

$\Rightarrow$ P(A) + P(B) + P (C) - P (A $\cap$ B)

- P (B $\cap$ C) - P (C $\cap$ A) = ${3 \over 8}$

$\therefore$ P(atleast one event occurs)

= P (A $\cup$ B $\cup$ 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)

= ${3 \over 8} + {1 \over {16}}$ = ${7 \over {16}}$