In a game you roll two fair dice. If the sum of the two numbers obtained is 3,4,9,10 or 11 you win $20. If the sum is 5,6,7 or 8 you pay $20. However if the scores on the dice are the same no one is required to pay.
(a) Construct a probability distribution table for the event
(b) Determine the expected outcome of this game. Is it fair?
(c) What should your winnings be for the game to be fair?
I will start it for you. Most numbers can be obtained in more than one way. 2 and 12, the only no pay numbers, can only be obtained one way.
3 = 2,1 or 1,2 = 2 ways
4 = 2,2 or 3,1 or 1,3 = 3 ways
5 = 3,2 or 2,3 or 1,4 or 4,1 = 4 ways
etc.
In order to answer these questions, we need to calculate the probabilities of each possible outcome when rolling two fair dice.
(a) Probability distribution table for the event:
First, let's list all possible outcomes when rolling two dice:
Dice 1 | Dice 2 | Sum
-------|--------|-----
1 | 1 | 2
1 | 2 | 3
1 | 3 | 4
1 | 4 | 5
1 | 5 | 6
1 | 6 | 7
2 | 1 | 3
2 | 2 | 4
2 | 3 | 5
2 | 4 | 6
2 | 5 | 7
2 | 6 | 8
3 | 1 | 4
3 | 2 | 5
3 | 3 | 6
3 | 4 | 7
3 | 5 | 8
3 | 6 | 9
4 | 1 | 5
4 | 2 | 6
4 | 3 | 7
4 | 4 | 8
4 | 5 | 9
4 | 6 | 10
5 | 1 | 6
5 | 2 | 7
5 | 3 | 8
5 | 4 | 9
5 | 5 | 10
5 | 6 | 11
6 | 1 | 7
6 | 2 | 8
6 | 3 | 9
6 | 4 | 10
6 | 5 | 11
6 | 6 | 12
Now we can calculate the probability of each sum occurring:
Sum | Frequency | Probability
-----|-----------|------------
3 | 1 | 1/36
4 | 2 | 2/36
5 | 4 | 4/36
6 | 6 | 6/36
7 | 6 | 6/36
8 | 4 | 4/36
9 | 3 | 3/36
10 | 2 | 2/36
11 | 1 | 1/36
(b) Expected outcome of the game and fairness assessment:
To determine the expected outcome, we multiply each outcome by its respective probability and sum them up:
Expected outcome = (3 * 1/36 + 4 * 2/36 + 9 * 3/36 + 10 * 2/36 + 11 * 1/36) - (5 * 4/36 + 6 * 6/36 + 7 * 6/36 + 8 * 4/36) = (3 + 8/9 + 27/12 + 10/9 + 11/36) - (10/9 + 4/3 + 7/3 + 8/9) = 107/36 - 79/9 ≈ -$0.03
The expected outcome is approximately -$0.03, which means that on average, you would lose about $0.03 per game. Since the expected outcome is negative, the game is not fair.
(c) To make the game fair, the expected outcome should be zero. So, to find the fair winnings, we need to determine the amount that would result in an expected outcome of zero.
Let's call the fair winnings "x":
Expected outcome = (3 * 1/36 + 4 * 2/36 + 9 * 3/36 + 10 * 2/36 + 11 * 1/36) * x - (5 * 4/36 + 6 * 6/36 + 7 * 6/36 + 8 * 4/36) * x
Simplifying the equation:
107/36 * x - 79/9 * x = 0
Multiplying both sides by 36/107:
x - (79/9) * (36/107) * x = 0
x - 2844/963 = 0
x = 2844/963 ≈ $2.95
To make the game fair, the winnings should be approximately $2.95.