A Las Vegas roulette wheel has 38 numbered slots. Roulette wheels are calibrated

Question

A Las Vegas roulette wheel has 38 numbered slots. Roulette wheels are calibrated such that each outcome is

equally likely. To play roulette, you choose a number; you win$35 if the ball ends up in that slot and lose $1 otherwise.

a) Draw a box model appropriate for this problem.

b) You play five times. Find the probability that you will not lose money.

c) About how much do you expect to win (or lose) in 100 plays if you make this bet on each play?

d) Estimate the chance that you will win more than $12 in 100 plays.

e) Calculate the chance that you win 4 times in 100 plays. (Hint: You might need another box model)

f) Estimate the chance that you will win more than 4 times in 100 plays.

Explanation / Answer

This is a binomial distribution with the following probabilities

•The probability of success p(s) = 1/38

•The probability of failure p(f) = 37/38

c) Expected win in each game is p(s) * 35 + p(f) * -1 = -0.05295. So, in 100 games the expected loss is 100*-0.05295 = 5.295

e) Probability of winning exactly 4 times is P(X=4)
where P(X=n) for 100 plays is 100Cn * p(s)n * p(f)(100-n)
Substituting the values of n=4, we get the P(X=4) = 0.14534

f) Probability of winning more than 4 times in 100 plays is

P(X>4) = 1 – [P(X=4)+P(X=3)+P(X=2)+P(X=1)+P(X=0)]

Where P(X=n) for 100 plays is 100Cn * p(s)n * p(f)(100-n)

Substituting this we get P(X>4) = 1 – [0.069+0.188+0.251+0.222+0.145] = 0.124

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