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,

Question

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

TestHub · Accepted Answer

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 $D = 0$

$\Rightarrow α^{2} - 4 β = 0 \Rightarrow α^{2} = 4 β$

$\Rightarrow (α = 0, β = 0)$ or $(α = 2, β = 1)$

Thus, the required probability

$= {}^{2}C_{0} {(\frac{2}{3})}^{0} {(\frac{1}{3})}^{2} . {}^{2}C_{0} {(\frac{1}{3})}^{0} {(\frac{2}{3})}^{2} + {}^{2}C_{2} {(\frac{2}{3})}^{2} {(\frac{1}{3})}^{0} . {}^{2}C_{1} {(\frac{1}{3})}^{1} {(\frac{2}{3})}^{1}$

$= \frac{1}{9} \times \frac{4}{9} + \frac{4}{9} \times \frac{4}{9} = \frac{20}{81}$

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