Guess the total, win the total
A fair 10-sided die (faces 1 to 10) and a fair 6-sided die are rolled. Before the roll, you name a number. If it equals the sum of the two dice, you receive that many dollars; otherwise you get nothing.
What is the expected payout of the best guess?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
For each possible sum , the payout is .
Which sums are most likely? Among those, which pays most?
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. The value of guessing is times the probability of . The most likely sums are a whole range, and among them you should take the largest.
Step 1: the distribution. There are 60 equally likely outcomes. For a sum from 7 to 11, every value of the 6-sided die can be completed by the 10-sided die, so each of these sums has probability . Outside that range the probability drops by per step.
Step 2: compare.
- Guessing 11: .
- Guessing 12: .
- Guessing 13: , and larger sums are worse still.
Step 3. The best guess is 11, worth dollars.
The trap. Picking the “middle” sum, 8 or 9: it is no more likely than 11 and pays less. When the prize grows with the guess, take the largest of the equally likely values, and check one step beyond the plateau.