Let C_1 and C_2 be two biased coins such that the probabilities of getting head in a single toss are 2/3 and 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 the probability that the roots of the quadratic polynomial x^2-αx+β are real and equal, is
(A) 40/81
(B) 20/81
(C) 1/2
(D) 1/4
Answers
Given : probability of getting head in single toss of C₁ , P(H) = 2/3
probability of getting head in single toss of C₂, p(H) = 1/3
α is the no of heads that appear when C₁ is tossed twice and β is the no of heads that appear when C₂ is tossed twice.
To find : The probability that the roots of quadratic equation x² - αx + β are real and equal.
solution : as probability of getting head in single toss of C₁ is 2/3 so probability of getting no head in single toss would be 1/3
now coin is tossed twice,
no of heads = 0, probability = (1/3)² = 1/9
no of heads = 1 , probability = (2/3)² = 4/9
no of heads = 2, probability = (2/3)² = 4/9
similarly, probability of getting head in single toss of C₂ is 1/3 so probability of getting no head in single toss would be 2/3
now coin is tossed twice,
no of heads = 0, probability = (2/3)² = 4/9
no of heads = 1 , probability = (2/3)² = 4/9
no of heads = 2, probability = (1/3)² = 1/9
for real and equal roots , Discriminant = α² - 4β = 0
⇒α² = 4β
it is possible into two ordered pairs of (α, β) = (0, 0) and (2, 1)
so the probability = 1/9 × 4/9 + 4/9 × 4/9 = 20/81
Therefore the correct option is (B)