Question

One of the most intense moments in football (or soccer as you may know it) is a penalty. A penalty is awarded when the opposing team fouls one of your players in- side the ”box”, i.e. a rectangle around the goal. Your team chooses a player to take a direct free kick from 11 meters (12 yards). If the player makes it, the team scores. Otherwise, if the player fails, the scoring opportunity is wasted. For the following questions, assume that each penalty shoot is independent of each other and that the historical success rate of the player is 75%. Projecting the player’s historical success rate into the future, calculate the following probabilities:

i) out of 10 penalties, scores all of them
ii) out of 20 penalties, scores at least 18 out of 20 penalties
iii) is the assumption of independence justified in this sports context?

Answer 1

I)
P(X=x) = nCx * p^x * q^(n-x)  
P = 75.00%
n = 10

P(X=10)
= BINOM.DIST(10,10,0.75,FALSE)
= 0.056313515

II)  
P(X=x) = nCx * p^x * q^(n-x)  
P=   75.00%
n=   20

P(X>=18)
= 1-P(X<18)
= 1-P(X<=17)
= 1-BINOM.DIST(17,20,0.75,TRUE)
= 0.0913

III)
Yes, The assumption of independence is jusitified in sports context. each player has indepednent chance of goal. The player 1 is independent of scoring as compared to player 2.