Four coins and one second chance
Four fair coins are tossed, and you are paid 1 dollar for each head. After seeing the result, you may pick one coin showing tails and toss it again, once.
What is your expected payout with the best play?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
When is the second chance worth anything?
If there is at least one tail, re-tossing it adds half a head on average.
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. The extra toss is worth half a dollar whenever there is a tail to re-toss, and nothing when all four coins already show heads.
Step 1. Without the extra toss, the expected payout is .
Step 2. If at least one coin shows tails (probability ), re-tossing it turns it into a head with probability : worth dollar. Re-tossing is never worse, so you always do it when you can.
Step 3.
Variants. If you may re-toss every tail once, each coin ends heads with probability and the game is worth 3. If you may keep re-tossing tails forever, it is worth 4.