Determining Probability from Expectation

Question

A gambler is playing with a crooked(unfair) twelve-side die. The chances that certain side of a die will turn up is given in the table below, except for P5 and P6. Find P5 and P6 if it known that expectation is 4.5625.

 

Outcome     Probability
1                1/48
2                5/48
3                11/48
4                15/48
5                P5
6                P6
7                2/48
8                2/48
9                1/48
10              1/48
11              1/48
12              1/48

Answer 1

The probabilities sum to 1 so:

 

p₅ + p₆ = 8/48

 

The expectation is E(X) = 4.5625 = 219/48 so

 

(1)(1/48) + (2)(5/48) + ... + (12)(1/48) = 219/48

 

5p₅ + 6p₆ =  (219 - 176)/48 = 43/48

 

subtract 5 times the first equation:

 

p₆ = 43/48 - 40/48 = 3/48

 

p₅ = 8/48 - p₆ = 5/48