Statistics and Probability: JEE Advanced 2020 Mathematics 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