Answer:
a) The expected value of a die is 3.5
b) The fair value of this game is 4.25 dollars
c) The fair value of this game is 4.6667 dollars
Step-by-step explanation:
a) All the values are equally likely, so to get the expected value of the die we just sum all possible values and divide by the total amount of possible values, 6.
Thus, the expected value of a die is 3.5.
b) The fair value is the expected value of the game. You will reroll the dice only if you expect to get more value on the second throw than what did you obtain on the first one. Since the expected value of a die is 3.5, then you will only reroll the die if you obtain 3 or less, and the probability for that is 1/2.
This means that there is a probability of 1/2 to make a second roll, with expected value 3.5, and a probability of 1/6 to obtain each of 4, 5 or 6 with the first die and keep that value (because it is higher than the expected value from one dice). Therefore, the expected value of this game is
1/2 * 3.5 + 1/6 (4+5+6) = 4.25
The fair value of the game is 4.25 dollars.
c) You can interpret this game as if after your first roll, you can keep that value or play the game described in b, where you roll once and reroll once if you want. Thus, if you decide to decline the value of the first die and reroll, then the expected value of the game will be 4.25. As a result, you will only reroll if you obtain a 4 or lower in the first die, which happens with a probability of 4/6 = 2/3. If you obtain 5 or 6, which happen each case with probability 1/6, you will keep that value because it exceeds your expectations. As a consecuence, the expected value is
2/3 * 4.25 + 1/6*(5+6) = 14/3 = 4.6667
The fair value of the game is 14/3 = 4.6667 dollars.
