Question

A and b are equally good tennis players

Answer 1

Answer on Question #64896 – Math – Statistics and Probability

Question

A and B are equally good tennis players. Which of the following two events is more probable?

1) A beats B exactly in 3 games out of 4.

2) A beats B exactly in 5 games out of 8.

Solution

We need to use binomial distributions with $p = \frac{1}{2}, n = 4$ and $p = \frac{1}{2}, n = 8$.

1)

$$P(X = 3) = \frac{4!}{3!(4 - 3)!} \left(\frac{1}{2}\right)^3 \left(\frac{1}{2}\right)^{4 - 3} = 4 \left(\frac{1}{2}\right)^4 = \frac{1}{4}.$$

2)

$$P(Y = 5) = \frac{8!}{5!(8 - 5)!} \left(\frac{1}{2}\right)^5 \left(\frac{1}{2}\right)^{8 - 5} = \frac{8 \cdot 7 \cdot 6}{3 \cdot 2} \left(\frac{1}{2}\right)^8 = 56 \left(\frac{1}{2}\right)^8 = \frac{7}{32}.$$

The probability of $1^{st}$ event can be expressed as

$$P(X = 3) = \frac{1}{4} \cdot \frac{8}{8} = \frac{8}{32}.$$

Therefore, the first event is more probable than the second, because

$$P(X = 3) = \frac{8}{32} > P(Y = 5) = \frac{7}{32}.$$

**Answer**: A beats B exactly in 3 games out of 4.

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