1. Tardigrade
  2. Question
  3. Mathematics
  4. Let C1 and C2 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 C1 is tossed twice, independently, and suppose β is the number of heads that appear when C2 is tossed twice, independently. Then the probability that the roots of the quadratic polynomial x2-α x+β are real and equal, is

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Q. 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

JEE AdvancedJEE Advanced 2020

Solution:

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