91Ó°ÊÓ

The given statement is:

In the game of soccer, a penalty kick is a direct free-kick, taken from 12 yards out from the goal on the penalty mark. According to the article "Penalty Kicks in Soccer: An Empirical Analysis of Shooting Strategies and Goalkeeper's Preferences" (Soccer & Society,Vol. 10, No. 2. pp. 183-191) by M. Bar-Eli and O. Azar, 85% of penalty kicks placed by professional soccer players are successful.

$$\begin{gathered} n=15,p=0.85,\mathrm{and}x=15 \\ P\left(X=x\right)=\left(\begin{array}{c}n\\ x\end{array}\right)p^x{\left(1-p\right)}^{n-x} \\ P\left(X=15\right)=\left(\begin{array}{c}15\\ 15\end{array}\right){\left(0.85\right)}^{15}{\left(1-0.85\right)}^{15-15} \\ =\left(1\right){\left(0.85\right)}^{15}{\left(0.15\right)}^0 \\ =\left(1\right)\left(0.0874\right)\left(1\right) \\ =0.0874 \end{gathered}$$

$$\begin{gathered} P\left(Xâ‰\yen 13\right)=P\left(X=13\right)+P\left(X=14\right)+P\left(X=15\right) \\ P\left(X=13\right)=\left(\begin{array}{c}15\\ 13\end{array}\right){\left(0.85\right)}^{13}{\left(0.15\right)}^2 \\ =\left(105\right)\left(0.1209\right)\left(0.0225\right) \\ =0.2856 \\ P\left(X=14\right)=\left(\begin{array}{c}15\\ 14\end{array}\right){\left(0.85\right)}^{14}{\left(0.15\right)}^1 \\ =\left(15\right)\left(0.1028\right)\left(0.15\right) \\ =0.2312 \\ P\left(X=15\right)=\left(\begin{array}{c}15\\ 15\end{array}\right){\left(0.85\right)}^{15}{\left(0.15\right)}^0 \\ =\left(105\right){\left(0.85\right)}^{15}{\left(0.15\right)}^0 \\ =0.0874 \\ P\left(Xâ‰\yen 13\right)=P\left(X=13\right)+P\left(X=14\right)+P\left(X=15\right) \\ =0.2856+0.2312+0.0874 \\ =0.6042 \end{gathered}$$

The formula for mean is :

$\mathrm{μ}=np$

Substitute the values,

$$\begin{gathered} =\left(15\right)\left(0.85\right) \\ =12.75 \end{gathered}$$

Professional soccer players make an average of 12.5 penalty kicks every game.

The formula for standard deviation is :

$\mathrm{σ}=\sqrt{npq}$

Substitute the values,

$$\begin{gathered} \mathrm{σ}=\sqrt{\left(15\right)\left(0.85\right)\left(1-0.85\right)} \\ =\sqrt{1.9125} \\ =1.3829 \end{gathered}$$

In general, the average number of successful penalty kicks is 1.38 out of a total of 12.75.