JEE Advanced · 2020 · 27 Sep 2020 · Paper 1 · Probability XII

Let

C_{1}

and

C_{2}

be two biased coins such that the probabilities of getting head in a single toss are

\frac{2}{3}

and

\frac{1}{3}

, respectively. Suppose

α

is the number of heads that appear when

C_{1}

is tossed twice, independently, and suppose

β

is the number of heads that appear when

C_{2}

is tossed twice, independently. Then probability that the roots of the quadratic polynomial

x^{2} - α x + β

are real and equal, is

Answer: (B)

Solution

For coin

C_{1}

with

P (H) = \frac{2}{3}

, when tossed twice:

$P (α = 0) = \frac{1}{9}$ , $P (α = 1) = \frac{4}{9}$ , $P (α = 2) = \frac{4}{9}$

For coin

C_{2}

with

P (H) = \frac{1}{3}

, when tossed twice:

$P (β = 0) = \frac{4}{9}$ , $P (β = 1) = \frac{4}{9}$ , $P (β = 2) = \frac{1}{9}$

For real and equal roots of

x^{2} - α x + β

, we need

α^{2} = 4 β

.

This occurs when

(α, β) = (0, 0)

(2, 1)

.

Probability

= P (α = 0, β = 0) + P (α = 2, β = 1) = \frac{1}{9} \times \frac{4}{9} + \frac{4}{9} \times \frac{4}{9} = \frac{4}{81} + \frac{16}{81} = \frac{20}{81}

JEE Advanced · 2020 · 27 Sep 2020 · Paper 1 · Single correct option