JEE Advanced 2020 Paper 1Question 1
Let $C_1$ and $C_2$ be two biased coins such that the probabilities of getting head in a single toss are $\frac23$ and $\frac13$, 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
Probability
Solution
Equal roots need $\alpha^2=4\beta$: $(\alpha,\beta)=(0,0)$ with $\frac19\cdot\frac49=\frac4{81}$, or $(2,1)$ with $\frac49\cdot\frac49=\frac{16}{81}$. Total $\frac{20}{81}$.