Question

Task 1: Roulette wheel simulation

A roulette wheel has 38 slots of which 18 are red, 18 are black, and 2 are green. If a ball spun on to the wheel stops on the color a player bets, the player wins. Consider a player betting on red. Winning streaks follow a Geometric(p = 20/38) distribution in which we look for the number of red spins in a row until the first black or green. Use the derivation of the Geometric distribution from the Bernoulli distribution to simulate the game. Namely, generate Bernoulli(p = 20/38) random variates (0 = red; 1 = black or green) until a black or green occurs.

Code set-up

A while loop allows us to count the number of spins until a loss. If we use indicator variable lose to note a win (1) or loss (0), the syntax is “while we have not lost (i.e., lose==0), keep spinning.” Once you win, the while loop ends and the variable streak has counted the number of spins. Try running a few times.

streak = 0
lose = 0
p = 20/38
while(lose==0){
lose = (runif(1) < p) # generate Bernoulli with probability p
streak = streak + 1 # tally streak
}
streak

## [1] 2

The problem

The code chunk above performs the experiment once: spin the roulette wheel until you lose and record the number of spins. Simulate 1000 experiments. As usual, do this by wrapping the code chunk above within a for-loop and storing the number of spins streak in a vector.

# [Place code here]

Report the following:

• Histogram of the win streak length. Note that this is a discrete distribution so should place histogram bars at discrete values {0, 1, 2, …}. This may be done with the breaks option within hist. If your storage variable is called winstreak:

hist(winstreak, br=seq(min(winstreak)-0.5, max(winstreak+0.5)), main="")

• Average length of the win streak. [Answer here]
• Standard deviation of the winning streak lengths. [Answer here]
• Compare the empirical average and standard deviation in the previous two bullets to the true values from the Geometric(p = 20/38) distribution.

[Answer here]

• Longest winning streak. [Answer here]

Answer 1

Let X be the random variable indicating the win streak length. We know that X has a Geometric distribution with parameter, probability of success (the probability of a black or green spin) p=20/38

The expected value if X is

The standard deviation of X is

These are the theoretical average and standard deviation of win streak length

R code already given for task 1

-----

#set the random seed
set.seed(123)

streak = 0
lose = 0
p = 20/38
while(lose==0){
lose = (runif(1) < p) # generate Bernoulli with probability p
streak = streak + 1 # tally streak
}
streak
----

get this

This means we won 0 times betting on red, before we lost.

Note: This will not be same as "## [1] 2" a streak length of 2 given in the question as each run of the simulation gives different result, unless we set same the random seed

The problem

R code with comments

--

#set the random seed
set.seed(123)

#set the number of experiments
n<-1000
#initialize streak
winstreak<- numeric(n)

for (i in 1:n){
   streak = 0
   lose = 0
   p = 20/38
   while(lose==0){
       lose = (runif(1) < p) # generate Bernoulli with probability p
       streak = streak + 1 # tally streak
   }
   winstreak[i]<-streak
}
#Histogram of the win streak length.
hist(winstreak, br=seq(min(winstreak)-0.5, max(winstreak+0.5)), main="Histogram of the win streak length",xlab="win streak length")
#Average length of the win streak.
sprintf('Average length of the win streak is %.4f',mean(winstreak))
#Standard deviation of the winning streak lengths.
sprintf('Standard deviation of the win streak lengths is %.4f',sd(winstreak))
#Compare the empirical average and standard deviation with
#Geometric(p = 20/38) distribution.
sprintf('Theoretical average length of the win streak is %.4f',1/p)
#Standard deviation of the winning streak lengths.
sprintf('Theoretical Standard deviation of the win streak lengths is %.4f',sqrt((1-p)/p^2))
Longest winning streak.
sprintf('Longest winning streak is %g',max(winstreak))
--

get this

ans:

• Histogram of the win streak length.

• Average length of the win streak.  1.9010
• Standard deviation of the winning streak lengths 1.2903
• Compare the empirical average and standard deviation in the previous two bullets to the true values from the Geometric(p = 20/38) distribution.

We can see that the true value of Average length of the win streak is 1.90 and the empirical value of 1.9010 is close enough to it.

The true value of Standard deviation of the winning streak lengths is 1.3077 and the empirical value of 1.2903 is close enough to it

• Longest winning streak is 10