Probability

Question

In the game Chuck-a-Luck, three dice are rolled. A player bets a certain amount (says $1.00) on a number from 1 to 6. If the number appears on 1 die, the person wins $1.00. If it appears on 2 dice, the person wins $2.00, and if it appears on all 3 dice, the person wins $3.00. What are the chances of winning $1.00? $2.00? and $3.00?

Answer 1

The chances of winning $1 is when the number occurs on only one of the die.  The probability it occurs on one die is 1/6 so you want P(X = 1) out of a sample of n = 3 die.  This is a binomial event which has probability P(X = k) = (n C k) p^k (1-p)^k =  P(X = 1) = (3 C 1) (1/6)^1 (5/6)^2 = 0.3472222.  It's a binomial event since the probability of your number coming up on each die is constant (1/6) and you have a fixed sample (3 die).  The event that you roll your number on each die is binary (it either comes up or not).

 

Similarly, if they win $2 this means out of any 2 die, their number occurred in both cases which is P(X = 2) = (n C k) p^k (1-p)^k = P(X = 2) = (3 C 2) (1/6)^2 (5/6)^1 = 0.06944444.

 

Finally, if they win $3 this means out of any 3 die, their number occurred in all cases which is P(X = 3) = (n C k) p^k (1-p)^k = P(X = 3) = (3 C 3) (1/6)^3 (5/6)^0 = 0.00462963.