Answer:
You can expect to make $250.
Step-by-step explanation:
Possible outcomes:
For the pair of dice:
(1,1), (1,2), (1,3), (1,4), (1,5), (1,6)
(2,1), (2,2), (2,3), (2,4), (2,5), (2,6)
(3,1), (3,2), (3,3), (3,4), (3,5), (3,6)
(4,1), (4,2), (4,3), (4,4), (4,5), (4,6)
(5,1), (5,2), (5,3), (5,4), (5,5), (5,6)
(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)
So 36 total outcomes.
Probability of the second number rolled being the same as the first number rolled:
6 outcomes: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6)
Out of 36, thus:
![p = \frac{6}{36} = \frac{1}{6}](https://tex.z-dn.net/?f=p%20%3D%20%5Cfrac%7B6%7D%7B36%7D%20%3D%20%5Cfrac%7B1%7D%7B6%7D)
Expected value of 1 game:
1/6 probability of earning $25.
5/6 probability of losing $2.
Thus:
![E = 25\frac{1}{6} - 2\frac{5}{6} = \frac{25 - 10}{6} = 2.5](https://tex.z-dn.net/?f=E%20%3D%2025%5Cfrac%7B1%7D%7B6%7D%20-%202%5Cfrac%7B5%7D%7B6%7D%20%3D%20%5Cfrac%7B25%20-%2010%7D%7B6%7D%20%3D%202.5)
100 games:
100*2.5 = 250
You can expect to make $250.