JEE Main 2017 — Probability Question with Solution

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

P (exactly one of A or B occurs)

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

P (Exactly one of B or C occurs)

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

P (Exactly one of C or A occurs)

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

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

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

$$\therefore$$ 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)

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