The other day I was walking on a street, along which there are a lot of booths where people play games to gamble. I stopped in front of one booth. The host was warm and we started to talk.
“How to play?”, I asked.
“Well, simple.” He explained, “You pay $10 to play once. Then you toss a coin until you get a ‘head’ and the game is over. If you get a ‘head’ in the first toss, then I will pay you $1; if in the 2nd toss, I will pay you $2; 3rd I will pay you $4; … and nth I will pay 2^(n-1) dollars.”
“Interesting!” I then started to calculate if this game is fair to me. I needed to calculate the expected return from this game. If it’s larger than $10, then I win; otherwise I will lose. So what is the expected return?
The probability to get ‘head’ in the first toss is 1/2, 2nd time 1/4 and so on. So the expected return is 1×1/2 + 2×1/4 + 4×1/8 + … = 1/2 + 1/2 + 1/2 + 1/2 + … and the sum is infinity!
My return is infinity! And it’s much larger than $10 dollars. For sure I will play. I might become a millionaire today. I count my money in my wallet and I have $100.
“I am in!”, I told the host.
“Good decision.”, He similed.
Then I started to play. The first game I was unlucky. I got the head in the first toss. But it’s the risk I have to take to become a millionaire. So I keep playing.
Before I knew I already spent all my $100! I only won about $50. But at this point nothing could stop me from becoming a millionaire. So I keep playing until I lost all my money.
I was very disappointed, but I was more confused. The expected return is infinity, but why did I lose money? Did I make a mistake in the calculation?