Unlike other articles, this article deals with math. If you can't stand math, then please by all means, skip this article. Otherwise, this article is meant for you to enjoy.
A) The St. Petersburg Paradox. (The Coin Flipping Paradox)
We start with $1 on the table. I have a coin. If I flip a heads, the money on the table doubles. If I flip a tails, the game ends. If I keep flipping heads, then the earnings would keep doubling: $1, $2, $4, $8, $16, $32 .... At the end of the game, you may collect the amount of money on the table. The question is, how much should you pay to play this game?
Worded differently, the question is asking what is the expected value of earnings of the game? So, how much can you expect to win? Well, there is a 1/2 chance of winning $2 (1 heads, doubled 1 dollar once), 1/4 chance of winning $4 (2heads, doubled 1 dollars 2 times). If this goes on, our expected value would be (1/2 chance of winning $2) + (1/4 chance of winning $4) + (1/8 chance of winning $8) and so on. Mathematically, this would be:
Or simply:
sorry for the big text. technical issue
However, this can be quite a problem because in real life; when we actually play this game (try it out!), we don't win a lot per game, the average is only a few bucks. Here's some actual trials I conducted with myself for some real live experimental probability:
H = Heads, T = Tails
Trial 1> T $1
Trial 2> HHT $4
Trial 3> HT $2
Trial 4> HT $2
Trial 5> T $1
Trial 6> HT $2
Trial 7> T $1
Trial 8> T $1
Trial 9> HHHHT $16
Trial 10> T $1
Trial 11> HHT $4
Trial 12> T $1
Trial 13> T $1
Trial 14> HHT $4
Trial 15> HHHT $8
Trial 16> T $1
Well, I cheated a bit to make the stats a bit more even. :D I made sure there was 16 Tails, and 16 Heads. There were 16 Tails because there was a trial for every tails. And there were 16 heads to counter balance the statistics of tails. Well, these stats are definitely extremely accurate to real world results. If you notice, the chance of rolling an infinite amount of heads is infinitely small. So, rolling an infinite number of heads is out of the question. The most I rolled was 4 heads (1/16 chance), so it was extremely unlikely that I would roll something of a more substantial quantity.
9 Heads would give me $512, but it's going to be a 1/512 chance in earning that 512 dollars. So playing this game is like trying your chances for the lotteries, testing your luck. This game is exactly like buying lottery tickets and back to our original question; how much should you pay so that it's fair when you play this game. The answer is definitely not infinity; we saw that the chance of rolling an infinite number of heads would be ridiculous. Get 10 quarters; roll all of it, let see you try to roll all heads. In fact, the exponential curve is so evident that it would be the same probability to flip around 21 heads to get a royal flush in poker.
So then, let's use the data obtained from my experiment instead of the mathematical equation to find the answer. We can simply do this buy finding the average of your earnings, or the middle quantity or your earnings. Either way, we will end up with a solution much closer to the truth. Let's try the average(mean) first:
Simple as it sounds: (1+4+2+2+1+2+1+1+16+1+4+1+1+4+8+1)/16 = 3.125
Now, let's try the median, or the answer in the middle: First I organized the numbers according to numerical value: 1 1 1 1 1 1 1 1 2 2 2 4 4 4 8 16. Then I picked out the middle two numbers and rounded it off, which was 1.5
So now we are left with two numbers, one $3.13 and $1.50. Both are outcomes of experimental probability. Of course, since $1.5o deals with more accurate numbers, $1.50 would be considered the answer for the question.
And the question I leave to you is simple, do you agree with the $1.50 as the amount you should pay to play this game, or would the solution be different? Why?
For more information on this topic, and other creative methods to exploit this problem, please visit: http://www.sas.upenn.edu/~baron/journal/9226/jdm9226.pdf
Thanks, and remember, your input is valued!
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