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Question 1

Ram and Shyam play table tennis with Ram's chance of winning a game being $$\frac{3}{5}$$ and Shyam's $$\frac{2}{5}$$. The winner gets $$1$$ point and loser $$0$$ points. The match terminates when one player has $$2$$ points more than the other. The probability of Ram winning the game at exactly the end of $$6^{th}$$ game, not before, is

For Ram to win exactly at the end of the sixth game, the first four games must leave the score tied, and Ram must then win the last two games. There are $$4$$ possible sequences for the first four games, each with probability $$\left(\frac{3}{5}\right)^2\left(\frac{2}{5}\right)^2$$. Thus the required probability is $$4\left(\frac{3}{5}\right)^4\left(\frac{2}{5}\right)^2=\frac{1296}{15625}$$.

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