- Home
- Standard 11
- Mathematics
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 probability that the roots of the quadratic polynomial $x^2-\alpha x+\beta$ are real and equal, is
$\frac{40}{81}$
$\frac{20}{81}$
$\frac{1}{2}$
$\frac{1}{4}$
Solution
$P ( H )=\frac{2}{3} \text { for } C _1$
$P ( H )=\frac{1}{3} \text { for } C _2$
for $C _1$
| No. of Heads $(\alpha)$ | $0$ | $1$ | $2$ |
| Probability | ${1}{9}$ | ${4}{9}$ | ${4}{9}$ |
for $C _2$
| No. of Heads $(\alpha)$ | $0$ | $1$ | $2$ |
| Probability | ${1}{9}$ | ${4}{9}$ | ${1}{9}$ |
for real and equal roots
$\alpha^2=4 \beta$
$(\alpha, \beta)=(0,0),(2,1)$
So, probability $=\frac{1}{9} \times \frac{4}{9}+\frac{4}{9} \times \frac{4}{9}=\frac{20}{81}$
