# Expected Value and Dice

Suppose you roll a die and you get whatever comes up (i.e. if I roll a 1 you get 1 dollar, if you roll a 6 you get 6 dollars, etc.). Also consider that you can roll the die once and get the money or you can choose to roll again and get whatever comes up the second time. Which option is better to choose?

So the expected value of the first option is $$\sum x p_x = 3.5$$ What is the expected value of the second option?

Would it be $$x-3.5$$

If your goal is to maximize the expected value, then you should reroll if and only if the first number you get is $3$ or lower, since the expected value of each die roll is $3.5$ as you pointed out. So you need to calculate the probability of getting $3$ or lower and $4$ or higher on the first roll and split into cases as to when you reroll, and compute the expectation of the second strategy this way. You should get that the expected value of the optimal strategy is $5/2 + 3.5/2 = 8.5/2$.