Problem 2.
You pay 5$ /round (nonrefundable) to play the game of rolling apair of fair dice.
If you roll an even sum, you lose, no pay off. If you roll anodd sum, that's your win (say, roll of 7 pays you 7$).
Discrete random variable X represents the winnings.
- For example, the lowest value of X is x=0 when you roll an evensum.
- For example, x=3 only when you roll {1,2} or {2,1}. You win1+2=3$ (odd sum). AndP ( x = 3 ) = P ( { 1 , 2 } ) + P ( { 2 , 1 }) = 1 / 36 + 1 / 36 = 1 / 18.
a) Find all possible values of X with theirprobabilities. Make the table as in Problem 1, a) for theprobability distribution of X. Above, we calculated just one row ofthe table:
| x | Add favorable dice Probabilities | P(x) |
| 0 | … | … |
| 3 | 1/36 + 1/36 | 1/18 |
| … | … | … |
Dice related probabilities are discussed in 5.1, page252, Example 5.
b) Find the expected value of X and interpretit.
Expected value is discussed in 6.1, page 320 (as mean),321 and Examples 5,6,7.
c) Does it make mathematical sense to play thegame? Remember, you have to pay 5$/game to play, what is your netgain/loss per game in the long run?
d) What price a (instead of 5$) would make thegame fair? It is called the fair price as you break even in thelong run: μ ( X ) − a = 0.
