Wimbledon finals

Serena Williams made a successful first serve 67% of the time in a Wimbledon finals match against her sister Venus. If she continues to serve at the same rate the next time they play and serves six times in the first game, determine the probability that:

1. all six first serves will be in.
2. at least four first serves will be in.

Correct answer:

p₆ =  0.0905
p₂ =  0.687

Step-by-step explanation:

$$\begin{gathered} p=67\%=\frac{67}{100}=0.67 \\ q=1-p=1-0.67=\frac{33}{100}=0.33 \\ n=6 \\ {p}_6=p^n=0.67^6=0.0905 \end{gathered}$$

$$\begin{gathered} C_4(6)=\left(\genfrac{}{}{0ex}{}{6}{4}\right)=\frac{6!}{4!(6-4)!}=\frac{6\cdot 5}{2\cdot 1}=15 \\ {C}_5(6)=\left(\genfrac{}{}{0ex}{}{6}{5}\right)=\frac{6!}{5!(6-5)!}=\frac{6}{1}=6 \\ OK:4,5,6 \\ {p}_4=\left(\genfrac{}{}{0ex}{}{n}{4}\right)\cdot p^4\cdot q^2=15\cdot 0.67^4\cdot 0.33^2\doteq 0.3292 \\ {p}_5=\left(\genfrac{}{}{0ex}{}{n}{5}\right)\cdot p^5\cdot q^1=6\cdot 0.67^5\cdot 0.33^1\doteq 0.2673 \\ {p}_2=p_4+p_5+p_6=0.3292+0.2673+0.0905=0.687 \end{gathered}$$

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