Infinity Game

Discrete Mathematics Level 2

You're invited to play a special coin-flipping game with a fairly simple rule: heads earn money; tails end party.

If you flip head at the first try, you'll be rewarded 1 dollar. If you flip another head at the second try, you'll be rewarded 2 more dollars. If you flip another consecutive head at the third try, you'll be rewarded 3 more dollars, and so on.

As long as the coin continuously turns up head, you can keep playing as many times as you like, but whenever it's tail face, the game is over immediately. (Therefore, it's also possible for you to gain nothing if you flip tail at your first try.)

Assuming the coin is unbiased, what is the expected earnings (in dollars) from this game?


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