Question

red die and a blue die are rolled. You win or lose money depending on the sum of the values of the two dice. If the sum is 5 or 9, you win $6. If the sum is 6, 7, or 12, you win $2. If the sum is any other value (2, 3, 4, 8, 10, or 11), you lose $4. Let X be a random variable that corresponds to your net winnings in dollars. What is the expected value of X

Answer 1

The expected value of X is $0.2222.

Explanation:

A probability is the number of desired outcomes divided by the number of total outcomes.

For each dice, there are 6 possible outcomes(values from 1 to 6). So for 2 dices, there are
`6^2 = 36` possible outcomes.

Desired outcomes:

Sum 5: (1,4), (2,3), (3,2), (4,1).

Sum 6: (1,5),(2,4),(3,3),(4,2),(5,1)

Sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)

Sum of 9: (3,6), (4,5), (5,4), (6,3);

Sum of 12: (6,6)

Probabilities:

4 + 4 = 8 outcomes with a sum of 5 or 9.

8/36 = 2/9 probability of a sum of 5 or 9.

12 outcomes with a sum of 6, 7 or 12.

12/36 = 1/3 probability of the sum being 6, 7 or 12.

36 - 20 = 16 outcomes in which the sum is any other value. So

16/36 = 4/9 probability of a sum of any other value.

What is the expected value of X?

2/9 probability of earning $6.

1/3 = 3/9 probability of earning $2.

4/9 probability of losing $4.

Multiplying each outcome by its probability, we find the expected value. So

`E = (2)/(9)*6 + (3)/(9)*2 - (4)/(9)*4 = (12 + 6 - 16)/(9) = (2)/(9) = 0.2222`

The expected value of X is $0.2222.