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
-----------------------------
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.
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.
