A couple of pieces of good news: First, in this post will be the answer to the problem I gave called Flipping Quarters; and second, I'll even work through the solution!
We have to figure out the average amount of money that the man pays you, minus $5. To start out, let's forget about the money and look at the actual coin flipping.
Since it's about as likely for the coin to give tails as it is to give heads, you would get heads 50% of the time. So on your 1st flip, there's a 50% chance you'll get heads - a 50% chance that you'll have to stop. What is the chance that you'll have to stop immediately after the 2nd flip? There's a 50% chance that you will actually get to the 2nd flip, and a 50% chance that you'll flip heads on it. Multiply those numbers, and you get 25%. Look at the diagram to the right if you don't understand. Now for the 3rd flip: there's a 25% chance there will be one (because you got tails on the 2nd flip), and a 50% chance you'll get heads. Multiply those numbers, and you get 12.5%. This continues on forever, with the chances halving for every extra flip. The series goes: 50%, 25%, 12.5%, 6.25%, 3.125%, etc.
Now let's go back to the money. If you get heads on your first flip, you will stop flipping and receive $1. There's a 50% chance this will happen, so on average you get $1 times 50% = 50 cents.
If you get heads on your second flip, you receive $2. There is a 25% chance that this will happen, so on average you get $2 times 25% = 50 cents. Add this to the last payment, and you get an average of $1.00.
If you get heads on your third flip, you get $4. There is a 12.25% chance that this will happen, so on average you get $4 times 12.25% = 50 cents. Add this to the last payment, and you get $1.50.
Since there's an infinite number of flips, I'll stop here. Now notice that for every flip, you get an average of 50 cents. On average, there is an infinite number of flips per game, so the 50 cents average really adds up. To infinity. Subtracting $5 doesn't even make a dent in the amount of money you get (on average). Yes, on average, you will not only come out ahead, you will make an infinite amount of money.
It may sound unlikely, but if you think about it, you may understand why it's so. Let me point out something: "infinity" is the average amount of money you get per game, not the actual amount. This partly means you'd eventually play a game that never ended.
You might wonder why all the 50 cents added up. Here's my answer: look at the picture to the right. It represents the average games; the first section ($1) represents the games with 1 flip, the next section ($2) represents the games with 2 flips, the next ($4) represents the games with 3 flips, etc. Look at the $1 game. It only occurs 50% of the time, so it only takes up 50% of the square. Since it only occurs 50% of the time, the average amount of money it gives you is 50% of $1, which is 50 cents. Now continue with the remaining games. They all exist, so you add the money from all of them.
You might wonder why all the 50 cents added up. Here's my answer: look at the picture to the right. It represents the average games; the first section ($1) represents the games with 1 flip, the next section ($2) represents the games with 2 flips, the next ($4) represents the games with 3 flips, etc. Look at the $1 game. It only occurs 50% of the time, so it only takes up 50% of the square. Since it only occurs 50% of the time, the average amount of money it gives you is 50% of $1, which is 50 cents. Now continue with the remaining games. They all exist, so you add the money from all of them.
You wouldn't actually ever play an infinitely long game. Perhaps a better way of putting it would be: if you play long enough, eventually you would play a game that took more flips than any arbitrarily large number. On the other hand, if you played long enough, eventually you would play a game that took longer than your lifetime - although it would be very unlikely to start such a game during your lifetime.
ReplyDeleteThat's true; when I wrote that sentence, I was actually thinking of the limit as the number of tries approaches infinity. I talked about the game never ending to illustrate the point that games could get very long.
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