Matt Murray, the Pittsburgh Penguins goalie, has a 0.931 save percentage in the
ID: 3298337 • Letter: M
Question
Matt Murray, the Pittsburgh Penguins goalie, has a 0.931 save percentage in the playoffs. There are many (correct) ways to interpret this value, one of which is Matt Murray has a 0.931 chance of making a save on each shot he faces. Let's explore how rare (or not rare) it would be for Matt Murray to make 30 saves in a row on 30 total shots in a game. (a) Use RStudio: Simulate Matt Murray's save percentage over the course of 200 shots on goal. (b) Use RStudio: Simulate the outcomes (here, a save or not a save) of 30 shots faced in a row. What is his save percentage based on this sample of 30 shots? (c) Use RStudio: Simulate the process of Murray facing 30 shots in a row 50 times. How many times (out of the 50) did Murray make all 30 saves? What is the probability he stops all 30 shots? (d) Use RStudio: Simulate the process of Murray facing 30 shots in a row 500 times. How many times (out of the 500) did Murray make all 30 saves? What is the probability he stops all 30 shots? (e) Use RStudio: Simulate the process of Murray facing 30 shots in a row 5000 times. How many times (out of the 5000) did Murray make all 30 saves? What is the probability he stops all 30 shots? (f) Make a comment or two about the probabilities you computed in parts c-e. If Matt Murray faces 30 shots in his next playoff game, is it likely or unlikely he will save all 30? Explain. Does it appear the probability he will save all 30 converges? Explain. (g) Bonus: We just went through a bunch of RStudio simulation to calculate the probability Murray saves all 30 shots (i.e. he has a shutout) in his next playoff game. Suppose we wish to be more general. Supply a simple method we could use to find the probability Murray stops all shots (i.e. he has a shutout) in his next playoff game. Notice I didn't specify how many shots he would face (like we did above). What is this probability (go find the data)? What assumption is needed to make this probability calculation? Is this assumption reasonable? Explain.Explanation / Answer
The R-code is given below:
set.seed(154)
p=0.931
########### (A)
n=200
sim_1 <- rbinom(n,1,p)
#Matt Murray's percentage save over the course of 200 shots
sum(sim_1)/n
############
############ (B)
n=30
sim_2 <- rbinom(n,1,p)
#Matt Murray's percentage save over the course of 30 shots
sum(sim_2)/n
#############
############# (C)
n=50
sim_3=numeric(n)
for(i in 1:n)
sim_3[i]=sum(rbinom(30,1,p))
# Number of times Murray make all 30 saves
sum(sim_3==30)
# Probability of all stops
sum(sim_3==30)/n
##############
############## (D)
n=500
sim_4=numeric(n)
for(i in 1:n)
sim_4[i]=sum(rbinom(30,1,p))
# Number of times Murray make all 30 saves
sum(sim_4==30)
# Probability of all stops
sum(sim_4==30)/n
##############
############## (E)
n=5000
sim_5=numeric(n)
for(i in 1:n)
sim_5[i]=sum(rbinom(30,1,p))
# Number of times Murray make all 30 saves
sum(sim_5==30)
# Probability of all stops
sum(sim_5==30)/n
###############
############### (F)
#Actual probability that Murray stops all the shots
dbinom(30,30,p)
###############
For the three cases c-e, we get the probabilities as 0.12, 0.108 and 0.1124 while the actual probability is
0.1170821.
Thus we can see the the sequence is approaching to the actual probability as we increase n.
SInce the probability that he saves all the shots is very less so its an unlikely event.