What is the mathematical expectation for this game?

Question

Suppose you toss a coin and will win $5 if it comes up heads. If it comes up tails, you toss again. This time you will receive $10 if it comes up heads. If it comes up tails, toss again. This time you will receive $20 if it is heads and nothing if it comes up tails. What is the mathematical expectation for this game?

Answer 1

Good question.

First flip: Half the time it's heads and you win $5, so $5 *0.5 = $2.50

If tails, then heads, win $10, so $10 * 0.5 * 0.5 = $2.50

If tails, then tails, then heads, win $20, so $20 * 0.5 * 0.5 * 0.5 = $2.50

If tails, then tails, then tails, win nothing, so $0 * 0.5 * 0.5 * 0.5 = $0

We can check that all of those fractional parts, added together, make 1, or all possibilities.

Expected value is the sum of those dollar amount results, or $7.50.

So, if someone says you can play for $5, then play. If the game costs $10, then don't play.