Coin Flipping Puzzle: Choose Between 2 Games

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I have a puzzle which asks the following question:

You have the choice to pick between two games. Assume the coins are fair.

Game 1: You toss a coin once. Heads you win \$10 and tails you lose \$10.

Game 2: You toss a coin 10 times. Heads you win \$1 and tails you lose \$1.

Which game do you prefer?

My initial thoughts were that I would prefer the second game, as I would be wiped out a lot slower.

How would you approach this question?

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Which you prefer depends on your personality. If you are risk-averse, you should prefer the second game. If you are a gambler, you may prefer the first, with high risk and high reward. Even if you are generally risk-averse, there are circumstances under which you would prefer the second game. Look at the first paragraph of How to Gamble If You Must