A fair coin

2011-07-02 00:10:30

Now let’s play a game. I have a fair coin, which means the probability of showing heads or tails is equally 50%. The rule is very simple, if the number of showing heads is more than 60 in 100 trials, you will win \$40, otherwise you will win nothing. The price of the game is \$1, will you like to play?

Firstly it seems plausible that you would have a good chance to win. Well in fact, every time you play the game, the expected return is actually negative. Tossing a fair coin is a Bernoulli process, so basically you can calculate the probability that you would win:

$$P = \sum_{k=60}^{100} p^k (1-p)^{100-k}$$

By Central Limit Theorem, we know when the number of trials is large, we can approximate Bernoulli by normal distribution, with mean np and variation np(1-p). Thus:

$$P(x>60) \cong 1 - N(\frac{60-50}{5}) = 1 - N(2)$$

The probability is around 0.0227, so the expected return is $0.0227*40 - 1 = - 0.09 < 0$.

Even without calculating the probability, if you know about “3 sigma rules”, which is that for a normal distribution, 68.3% falls into “one sigma” interval, 95.4% falls into “two sigma”, 99.7% falls into “three sigma”, then you would know that the probability of falling in the right side of “two sigma” interval is less than $(1-95\%)/2 = 2.5\%$, thus the expected return is less than $40*2.5\%-1=0$, which also suggests the negative return.

This story tell us:

  • It’s unlikely you will win against the game/rule makers.
  • Trust statistics is at least better than pure intuition.