Question

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 $\alpha$ is the number of heads that appear when $C_1$ is tossed twice, independently, and suppose $\beta$ is the number of heads that appear when $C_2$ is tossed twice, independently. Then the probability that the roots of the quadratic polynomial $x^2-\alpha x+\beta$ are real and equal, is

• A. $\frac{40}{81}$
• B. $\frac{20}{81}$
• C. $\frac{1}{2}$
• D. $\frac{1}{4}$

Answer 1

Answer: (B)

For $C_{1}P(h)=\frac{2}{3}$, for $C_{2}P(h)=\frac{1}{3}$,
given roots of the equation $x^2-\alpha x+\beta =0$ equal real roots

So required probability $={\left(\frac{2}{3}\right)}^2\cdot {}^2{C}_1\cdot \frac{1}{3}\cdot \frac{2}{3}+{\left(\frac{1}{3}\right)}^2{\left(\frac{2}{3}\right)}^2$
$=\frac{16}{81}+\frac{4}{81}=\frac{20}{81}$