Expected Value: Coins

Calculate expected value for coin scenarios

Expected Winnings in a Coin Game

Win $10

Lose $5

Flip a fair coin. Heads wins $10, tails loses $5. What is your expected profit?

Let X = profit from one game.

Outcome	Profit (x)	P(x)
Heads	+$10
Tails	-$5

Answer

E(X) = +$2.50

On average, you profit $2.50 per game. This is a favorable bet!

A fair game has E(X) = 0. If heads pays $W and tails loses $5:

Fair game: Heads pays $5, tails loses $5.
Our game pays $10 for heads (double), making it favorable to the player.

By linearity of expectation, if you play n games:

10 games

+$25

100 games

+$250

1000 games

+$2,500

Casinos ensure E(X) < 0 for all their games so they profit in the long run.