Question

A professional soccer player succeeds in scoring a goal on 84% of his penalty kicks. assume that the success of each kick is independent. (a) in a series of games, what is the probability that the first time he fails to score a goal is on his fifth penalty kick? (b) what is the probability that he scores on 5 or fewer of his next 10 penalty kicks? (c) suppose that our soccer player is out of action with an injury for several weeks. when he returns, he only scores on 5 of his next 10 penalty kicks. is this evidence that his success rate is now less than 84%? explain.

Answer 1

Geometric Distributions:Deals with the number of trials required until a single success.
 Geometric random variable: the number of trials (y) it takes to get a success in a geometric setting.
 When do you use it? To find first success or failure
 Calculator Commands
 When it takes more than n trials...1-geometcdf(p,n)
 When it take n trials... geometpdf(p,x)
 When it it takes a max of n trials...geometcdf(p,x)
 a. geometcdf(.16,5) = .0797
 b. binomcdf(10,.84,5) = .013
 c. From par b. if the soccer player's succes rate were still 84%, the probability that he would score 5 or few goals in 10 penalty kicks is 0.0130. This is so low thatwe should be suspicious about whether he can still hit 84% of his shots. We have convincing that his penalty kick success rate has fallen below 84%

Answer 2

a. Using the geometric probability:  .84^4 * (1-.84) = 0.07966; this means that the probability that he makes his first * the probability he makes his second * the probability he makes his third * the probability he makes his fourth * the probability he misses his fifth. 

b. P (X<= 5) = binomcdf (10, 0.84, 5) = 0.0130

c. From the answer in B, if the success rate of the player still continues at 84%, the probability that he will score 5 or few goals in the 10 penalty kicks will still remain at 0.0130. This is quite low, we should examine him about whether he can still hit at 84% of his shots. So given from this data, we could say that his penalty kick success rate has fallen below the given rate of 84%.