Question

(3) In the game of Chuck-a-luck, a player places a $1 bet on a number from 1 to 6. Three dice are then rolled. The player wins $1 for each die with the number they bet upon on it.

a. Construct the probability distribution that includes all four events in this game.

b. What is the expected value of this game?

c. Instead of $1, how much should be charged to make this a fair game?

Answer 1

the probability that a die rolled shows the number the player bet upon is 1/6

Let X be the number of dies out of 3, which show the number the player bet upon.

X has a binomial distribution with parameters, number of trials n=3 and success probability p=1/6.

the probabilities for X=0,1,2,3 are

We have 4 possible outcomes/events

• X=0, No die has the number bet: Payoff for the player = -$1, with a probability 0.5787
• X=1, 1 die has the number bet: Payoff for the player = $0, with a probability 0.3472
• X=2, Two dice have the number bet: Payoff for the player = $1, with a probability 0.0694
• X=3, Three dice have the number bet: Payoff for the player = $2, with a probability 0.0046

a)Let Y be the monetary payoff for the player betting $1 on this game.  The probability distribution of the payoff to the player is

b) The expected value Y is

The expected value of the game is -$0.5

c) Let $m be charged in place of $1. Let the player wins $1 for each die with the number they bet upon on it.

The payoffs are

• X=0, the payoff is -$m
• X=1, the payoff is $(1-m)
• X=2, the payoff is $(2-m)
• X=3, the payoff is $(3-m)

with the probability distribution of the payoffs are

The expected value of this game is

To make this a fair game, the expected value of the game should be zero.

that is

ans: To make this a fair game, $0.50 should be charged, instead of $1.