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 − α x + β are real and equal, is
B. 20 81
For tossing two coins independently, maximum and minimum heads that can appear will be 2   &   0 respectively. Roots of equation x 2 − α x + β = 0 are real and equal, then discriminant D = 0 ⇒ α 2 − 4 β = 0 ⇒ α 2 = 4 β ⇒ α = 0 ,   β = 0 or α = 2 ,   β = 1 Thus, the required probability = C 0 2 2 3 0 1 3 2 . C 0 2 1 3 0 2 3 2 + C 2 2 2 3 2 1 3 0 . C 1 2 1 3 1 2 3 1 = 1 9 × 4 9 + 4 9 × 4 9 = 20 81
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