Question

You roll a pair of honest dice. If you roll a total of 7, you win $18; if you roll a total of 11, you win $54; if you roll any other total, you lose $9. Find the expected payoff for this game.

Answer 1

Answer: -1
The negative value indicates a loss

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Step-by-step explanation:

Define the three events
A = rolling a 7
B = rolling an 11
C = roll any other total (don't roll 7, don't roll 11)

There are 6 ways to roll a 7. They are
1+6 = 7
2+5 = 7
3+4 = 7
4+3 = 7
5+2 = 7
6+1 = 7
Use this to compute the probability of rolling a 7
P(A) = (number of ways to roll 7)/(number total rolls) = 6/36 = 1/6
Note: the 36 comes from 6*6 = 36 since there are 6 sides per die

There are only 2 ways to roll an 11. Those 2 ways are:
5+6 = 11
6+5 = 11
The probability for event B is P(B) = 2/36 = 1/18

Since there are 6 ways to roll a "7" and 2 ways to roll "11", there are 6+2 = 8 ways to roll either event.
This leaves 36-8 = 28 ways to roll anything else
P(C) = 28/36 = 7/9

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In summary so far,
P(A) = 1/6
P(B) = 1/18
P(C) = 7/9

The winnings for each event, let's call it W(X), represents the prize amounts.
Any losses are negative values
W(A) = amount of winnings if event A happens
W(B) = amount of winnings if event B happens
W(C) = amount of winnings if event C happens
W(A) = 18
W(B) = 54
W(C) = -9

Multiply the probability P(X) values with the corresponding W(X) values
P(A)*W(A) = (1/6)*(18) = 3
P(B)*W(B) = (1/18)*(54) = 3
P(C)*W(C) = (7/9)*(-9) = -7

Add up those results
3+3+(-7) = -1

The expected value for this game is -1.
The player is expected to lose on average 1 dollar per game played.

Note: because the expected value is not 0, this is not a fair game.

Answer 2

Remark
There are 2 ways that you can throw 11
5 6 and 6 5

There are 6 ways that you can throw a 7
1 6 6 1
2 5 5 2
3 4 4 3

So the total number of successes is 6 + 2 = 8. They are not even in their payout, but that's the number used to find how many ways you can lose.

Step one
How many ways can you throw 2 honest dice?
There are 6 numbers on each one
6 * 6 = 36

Step Two
How many ways can you win?
The remarks say that there are 8 ways to win.

Step Three
How many ways can you lose?
36 - 8 = 28

Step Four
What are the expectations for this game.
Expectations = 54 * (2/36) + 18*(6/36) - 9(28/36)
Expectations = 3 + 3 - 7
Expectations = - 1

It means every time you pick up the dice and throw them, you should expect to lose 1$ This is not a very good game to play, but it is better than over under 7 which is quite popular in county fairs.

The house always wins!