MHB Expected Value of Rolling a Pair of Dice - Fair Price to Play

[rymatson406]

We roll a pair of dice. If the sum of the dice is 7, you pay me $28. If the sum is not 7, I pay you the number of dollars indicated by the sum of the dice. What is the price that you should pay to play the game that would make the game fair?

 

[MarkFL]

I deleted the duplicate of this thread in the Advanced Probability subforum. We ask that you post a question only once, as doing otherwise can lead to duplication of effort on the part of our helpers. I'm sure you can understand that the time and effort of our helpers is valuable, and we don't want to see it wasted. :D

edit: I did the same with your other question.

 

[chisigma]

We roll a pair of dice. If the sum of the dice is 7, you pay me $28. If the sum is not 7, I pay you the number of dollars indicated by the sum of the dice. What is the price that you should pay to play the game that would make the game fair?

Among the 36 possible results of the roll of two dice, those whose sum is k are k-1 for k ranging from 2 to 7 and 12 - (k-1) for k ranging from 8 to 12. The expected win your opponent is ...

$\displaystyle E\ \{ O \} = \frac{(1 + 2 + 3 + 4 + 5)\ 14}{36} = 5.833... \text{dollars}$

Your expected win is...

$\displaystyle E\ \{ Y \} = \frac{28}{6} = 4.666... \text{dollars}$

... and it is lower. For a fair game You should ask 35 dollars...

Kind regards

$\chi$ $\sigma$