Hockey - game

A hockey goalkeeper has a save success rate of 93.5%. What is the probability that he will keep a clean sheet in a match when facing 25 shots? What is the probability that he will concede a maximum of 2 goals?

Correct answer:

p₀ =  0.1863
b =  0.7803

Step-by-step explanation:

$$\begin{gathered} q=93.5\%=\frac{93.5}{100}=0.935 \\ r=1-q=1-0.935=\frac{13}{200}=0.065 \\ n=25 \\ {p}_0=\left(\genfrac{}{}{0ex}{}{n}{0}\right)\cdot r^0\cdot (1-r{)}^{n-0}=(\genfrac{}{}{0ex}{}{25}{0})\cdot 0.065^0\cdot (1-0.065{)}^{25-0}=1\cdot 0.065^0\cdot 0.935^{25}=0.1863 \end{gathered}$$

$$\begin{gathered} p_1=\left(\genfrac{}{}{0ex}{}{n}{1}\right)\cdot r^1\cdot (1-r{)}^{n-1}=(\genfrac{}{}{0ex}{}{25}{1})\cdot 0.065^1\cdot (1-0.065{)}^{25-1}=25\cdot 0.065^1\cdot 0.935^{24}\doteq 0.3238 \\ {p}_2=\left(\genfrac{}{}{0ex}{}{n}{2}\right)\cdot r^2\cdot (1-r{)}^{n-2}=(\genfrac{}{}{0ex}{}{25}{2})\cdot 0.065^2\cdot (1-0.065{)}^{25-2}=300\cdot 0.065^2\cdot 0.935^{23}\doteq 0.2702 \\ b=p_0+p_1+p_2=0.1863+0.3238+0.2702=0.7803 \end{gathered}$$

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