Question

Penalty Kicks. 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. In 15 penalty kicks by professional soccer players, what is the probability that

(a) all are successful?

(b) at least 13 are successful?

(c) Find and interpret the mean and standard deviation of the number of successful penalty kicks out of 15.

Short answer

Part (a) 0.0874

Part (b) 0.6042

Part (c) mean is 12.75 and the standard deviation is 1.3829.

Step-by-step solution

Part (a) Step 1. Given information.

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.

Part (a) Step 2. Find the probability that all are successful.

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

Part (b) Step 1. Find the probability that at least 13 are successful.

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

Part (c) Step 1. Find the mean of the number of successful penalties kicks out of 15.

The formula for mean is :

$\mu =np$

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.

Part (c) Step 2. Find the standard deviation of the number of successful penalties kicks out of 15.

The formula for standard deviation is :

$\sigma =\sqrt{npq}$

Substitute the values,

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

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