Imagine you are playing a game at a casino. You give up some amount of money.

The dealer flips a (fair) coin. If it is heads, you win back your money double.

If it is tails, you get no money.

Since the game is 50% - 50%, you can't expect to just win money automatically. On average, if you bet $1 every time and play the game over and over, you can expect to break even.

However, consider this strategy:

Start here:

Bet $1. If you win, start over. If not, continue:

Bet $2. If you win, start over. If not, continue:

Bet $4. If you win, start over. If not, continue:

Bet $8. If you win, start over. If not, continue:

Bet $16. If you win, start over. If not, continue:

etc...

So - what happens? Eventually, you have to win. It is completely improbable that you would lose forever.

What happens if you win at step 5? You win back 16. What did you lose (steps 1-4)?

8 + 4 + 2 + 1 = 15

So you actually won an entire dollar.

If you win at step 6, you get $32, and in steps 1-5 how much did you lose? Only $31.

So it seems like this strategy is 100% fool-proof. You always win $1 more than whatever you lost in all the previous steps. And when you repeat this strategy over and over, you can win an endless amount of money.

What is the flaw in this strategy? There are two flaws that I know of, although they are more-or-less the same, so you would probably come up with both of them I think.

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Imagine you are playing a game at a casino. You give up some amount of money.

The dealer flips a (fair) coin. If it is heads, you win back your money double.

If it is tails, you get no money.

Since the game is 50% - 50%, you can't expect to just win money automatically. On average, if you bet $1 every time and play the game over and over, you can expect to break even.

However, consider this strategy:

Start here:

Bet $1. If you win, start over. If not, continue:

Bet $2. If you win, start over. If not, continue:

Bet $4. If you win, start over. If not, continue:

Bet $8. If you win, start over. If not, continue:

Bet $16. If you win, start over. If not, continue:

etc...

So - what happens? Eventually, you have to win. It is completely improbable that you would lose forever.

What happens if you win at step 5? You win back 16. What did you lose (steps 1-4)?

8 + 4 + 2 + 1 = 15

So you actually won an entire dollar.

If you win at step 6, you get $32, and in steps 1-5 how much did you lose? Only $31.

So it seems like this strategy is 100% fool-proof. You always win $1 more than whatever you lost in all the previous steps. And when you repeat this strategy over and over, you can win an endless amount of money.

What is the flaw in this strategy? There are two flaws that I know of, although they are more-or-less the same, so you would probably come up with both of them I think.

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