Seven pays, even costs
You roll two fair dice repeatedly. A total of 7 wins you 1 dollar and ends the game. An even total loses you 1 dollar and ends the game. Any other total (3, 5, 9 or 11) means you roll again.
Find the expected payout.
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
The rolls that continue the game change nothing. Only the roll that ends it matters.
Compare the probabilities of a 7 and of an even total on one roll.
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s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. The game ends at the first roll that is a 7 or even. Given that a roll ends the game, compare the chances of the two endings.
Step 1. On one roll, a 7 has probability , and an even total has probability (two dice of the same parity).
Step 2. Given that the game ends, it ends with a 7 with probability
Step 3.
The trap. Thinking the extra rolls on 3, 5, 9, 11 help you. They only delay the outcome; the odds of the ending stay 1 to 3 against you.