Double six for a hundred
You roll two fair dice. If both show a six, you win 100 dollars. If exactly one shows a six, you lose dollars. Otherwise you roll both dice again, and so on until the game ends.
For which value of is the game fair?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
The rolls with no six change nothing. Only the outcomes that end the game matter.
Out of 36 outcomes, how many have two sixes, and how many have exactly one?
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Rolls without a six are simply repeated, so the game ends with the first roll that contains a six. Compare the two ways it can end.
Step 1: the ending outcomes. Out of 36 equally likely outcomes, 1 is a double six and 10 have exactly one six (5 with the six on the first die, 5 on the second).
Step 2: the final result. Given that the game ends, the double six has probability and exactly one six .
Step 3: fair price.
The trap. Counting “one six” as 5 outcomes, as if the dice were ordered and only the first die counted, gives . With two dice, exactly one six can happen in 10 ways.