Question

A roulette wheel has 38 slots, 18 of which are black, 18 red, and 2 green. When the wheel is spun, the ball is equally likely to land in any of the 38 slots. Gamblers can place a number of different bets in roulette. One of the simplest wagers chooses red or black. A bet of $1 on red will pay off an additional dollar if the ball lands in a red slot. Otherwise, the player loses the original dollar. When gamblers bet on red or black, the two green slot belong to the house.

a. A gambler’s winnings on a $1 bet are either $1 or -$1. Give the probabilities of each of these outcomes. Find the mean and standard deviation of a gambler’s winnings on one such bet.

b. Explain briefly what the law of large numbers tells us about what will happen in the long run if the gambler makes a very large number of bets on red.

c. The central limit theorem tells us the approximate distribution of the gambler’s mean winnings in 50 bets. What is this distribution? Be sure to specify the values of all necessary parameters.

d. If a gambler makes 50 bets on red, find a range into which there is roughly a 95% chance that the gambler’s mean winnings will fall. Multiply this result by 50 to get the middle 95% of the distribution of the gambler’s winnings on nights when he places 50 bets.

e. What is the probability that the gambler will win money if he places 50 bets on red? (This is simply the probability that the mean winnings are greater than 0.)

f. The casino takes the other side of these bets. If 100,000 bets are placed on red in a week at the casino, what is the distribution of the mean winnings of gamblers on these 100,000 bets? What range covers the middle 95% of the distribution of mean winnings in 100,000 bets? Multiply by 100,000 to get the range of gambler’s losses. What is the approximate probability that the casino will lose money on 100,000 bets?

(Note that the gamblers’ losses are the casino’s winnings. Parts (d) and (e) indicate that an individual gambler has a pretty decent chance of winning money on any given night – which creates excitement and keeps people gambling. However, part (f) indicates that in the long run, the casino essentially always wins.)

Answer 1

(a)

Let X be the gambler’s winnings on a $1 bet.

P(X = 1) = 18/38 = 9/19

P(X = -1) = 20/38 = 10/19 (There will 18 black slots + 2 green slots = 20 slots where the gambler lose)

E(X) = (9/19) * 1 + (10/19) (-1) = -1/19 = -0.05263158

E(X²) = (9/19) * 1² + (10/19) (-1)² = 1

Var(X) = E(X²) - [E(X)]² = 1 - ( -1/19)²  = 0.99723

Standard deviation of winnings = = 0.998614

(b)

As, the expected value of winnings is negative, the law of large numbers tells that the gambler will lose money if the gambler makes a large number of bets on red.

(c)

The central limit theorem tells us the approximate distribution of the gambler’s mean winnings in 50 bets is Normal distribution.

By the 68-95-99.7 rule the range in which the mean winnings will fall 95% of the time is within 2 standard deviations from the mean.

For single bet, the range is (-0.05263158 - 2 * 0.998614, -0.05263158 + 2 * 0.998614) = (-2.04986, 1.944596)

For 50 bets, the range is (50 * -2.04986, 50 * 1.944596) = (-102.493, 97.2298)

(d)

Expected winnings for 50 bets = 50 * -0.05263158 = -2.631579

Variance of winnings for 50 bets = 50 * 0.99723 = 49.8615

Standard deviation = = 7.061268

Probability that the gambler will lose money if he makes 50 bets = Probability that the mean is less than 0

= P(Z < (0 - (-2.631579)) / 7.061268)

= P(Z < 0.3727)

= 0.6453

(e)

The distribution of the mean winnings of the gamblers on these bets is Normal distribution.

As, Gamblers’ losses are the casino’s winnings, the expected casino’s winnings on each bet is 0.05263158 with variance of 0.99723

For 100,000 bets, the mean is 100,000 * 0.05263158 = 5263.158 and variance 100,000 * 0.99723 = 99723

Standard deviation = = 315.7895

Range covers the middle 95% of mean winnings in 100,000 bets is,

(99723 - 2 * 315.7895, 99723 + 2 * 315.7895)

(99091.42 , 100354.6)