Question

7. [10 marks] Consider an experiment of rolling two regular (or fair) balanced six-sided dice. a) List out all the possible outcomes.

b) Define X as the amount you will win in the following game.

You will win $100 when a double (two identical numbers) is rolled; you will win $10 when an odd sum is rolled; and you will win $30 when other even sum is rolled, excluding doubles. Define X as the amount you will win in this game. Construct the probability distribution table of X c) Calculate E(X) and SD(X)

d) Provide an interpretation of E(X). If you have to put a bet (i.e. money) to play the game, what should this bet be so that the game is considered fair?

Note: A “fair” game is a game where no one side has an edge over the other.

Answer 1

When two regular (or fair) balanced six-sided dice is rolled.

a) The possible outcomes are:

{(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(1,4),(3,5),(3,6),

(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

b) Let X be the amount of winning in the given condition.

A person win $100 when a double (two identical numbers) is rolled.

ie when outcomes are {(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)}

Probability of winning $100 is 6/36

A person win $10 when an odd sum is rolled.

i.e outcomes are:

{(1,2),(1,4),(1,6),(2,1),(2,3),(2,5),(3,2),(1,4),(3,6),

(4,1),(4,3),(4,5),(5,2),(5,4),(5,6),(6,1),(6,3),(6,5)}

Probability of winning $10 is 18/36

Probability of winning $30 is 12/36=1/6

Therefore the probability distribution of X is given by:

X=x P(X=x)

100 6/36

10 18/36

30 12/36

c) The expected value of X is given by:

The standard deviation of X is given by:

d) The expected value of X is 17.78 that means the expected winning of person the given game is $17.78

e) If we have to put a bet (i.e. money) to play the game, the bet should be of $17.78 so that the game is considered fair.