Question

You meet a man named Raoul on the street outside of the Madrid Casino in Las Vegas. Raoul wants to know if you’ll play a game. You’ll roll 3 6-sided dice, take the mean, and win if the mean is greater than or equal to 4.

Another man named Mr. T approaches you and tells you he’d pity the fool who would play Raoul’s game. In Mr. T’s game, you instead roll 2 6-sided dice, take the mean, and win if the mean is greater than or equal to 4.

You computed that the mean of all possible attempts for Raoul and Mr. T’s games are both 3.5. However, the standard error of the mean is about 1.0 for Raoul’s game and 1.2 for Mr. T’s game.

a) Why is the standard error of the mean higher for Mr. T’s game?

b) What’s the probability you would win if you played Raoul’s game?

c) What’s the probability you would win if you played Mr. T’s game?

Answer 1

a)

We know that the standard error of the mean increases with decrease in sample size and vice-versa.

The standard error of the mean higher for Mr. T’s game is higher because the sample size for Mr. T’s game which is 2 is smaller than the Raoul’s game which is 3.

b)

Using Normal approximation to Binomial distribution,  probability you would win if you played Raoul’s game

= P(X 4)

= P(Z (4 - 3.5) / 1)

= P(Z 0.5)

= 0.3085 (Using Standard Normal Distribution tables)

c)

Using Normal approximation to Binomial distribution,  probability you would win if you played Mr. T’s game

= P(X 1.2)

= P(Z (4 - 3.5) / 1.2)

= P(Z 0.4167)

= 0.3384 (Using Standard Normal Distribution tables)