and
Let C1and C2 be two biased coins such that the probabilities of getting head in a single toss are 2/3 and 1/3
respectively. Suppose alpha is the number of heads that appear when C1 is tossed twice
independently, and suppose beta is the number of heads that appear when C2 is tossed twice,
independently. Then the probability that the roots of the quadratic polynomial x^2-ax+ b we
real and equal, is
(A) 40/81
(B) 20/81
(C) 1/2
(D) 1/4
Answers
Given : C1and C2 be two biased coins such that the probabilities of getting head in a single toss are 2/3 and 1/3 respectively. Suppose alpha is the number of heads that appear when C1 is tossed twice independently, and suppose beta is the number of heads that appear when C2 is tossed twice,
independently.
To Find: probability that the roots of the quadratic polynomial x²2-αx+ β be
real and equal, is
(A) 40/81
(B) 20/81
(C) 1/2
(D) 1/4
Solution:
alpha is the number of heads that appear when C1 is tossed twice
α P(α)
0 (1/3)(1/3) = 1/9
1 (2/3)(1/3) + (1/3)(2/3) = 4/9
2 (2/3)(2/3) = 4/9
β is the number of heads that appear when C2 is tossed twice,
β P(β)
0 (2/3)(2/3) = 4/9
1 (1/3)(2/3) + (2/3)(1/3) = 4/9
2 (1/3)(1/3) = 1/9
x² - αx + β = 0
Roots are real and Equal if
α² = 4β
only possible when α & β are zero or α = 2 & β = 1
α = 0 & β = 0 => (1/9 ). (4/9 ) = 4/81
α = 2 & β = 1 => (4/9)(4/9 ) = 16/81
4/81 + 16/81
= 20/81
probability that the roots of the quadratic polynomial x^2-ax+ b be
real and equal, is = 20/81
option B is correct
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