JEE Main · 2021 · 26 Aug 2021 · Shift 1 · Probability

Let

A

and

B

be independent events such that

P (A) = p, P (B) = 2 p

. The largest value of

p

, for which

P

(exactly one of

A, B

occurs)

= \frac{5}{9}

, is:

Answer: (A)

Solution

Independence gives

P (exactly one of A, B) = P (A) (1 - P (B)) + (1 - P (A)) P (B) = p (1 - 2 p) + (1 - p) (2 p) = 3 p - 4 p^{2} .

Setting

3 p - 4 p^{2} = \frac{5}{9}

gives

36 p^{2} - 27 p + 5 = 0

, whose roots are

p = \frac{27 \pm 3}{72}

, that is

p = \frac{5}{12}

and

p = \frac{1}{3}

.
Both satisfy

2 p \leq 1

, so both are admissible probabilities, and the larger of the two is

\frac{5}{12}

JEE Main · 2021 · 26 Aug 2021 · Shift 1 · Single correct option