Misreading the Law (of Averages)
I recently received the following letter from one of my readers, Dave Beck.
Don,
Scoblete's recent Casino City Times column about charting has me wondering.
If charting is useless, then the triangle-shaped diagram of the dice combination possibilities must be useless. In one breath he says the percentages are based on this fact of probabilities. But in another breath he says the possibilities don't make a difference. If I chart a craps table for two hours and the number 9 has not been rolled in that time frame (it could happen) my little pyramid chart screams that the 9 should 'catch up' at some point. It may take ten years, but it will catch up. Right or Wrong? So what is so bad about charting as a tool for guessing the probability that an event will randomly "catch up." Apples and Oranges?
Well Dave, thanks for your letter. This is a topic that needs to be discussed and understood. As to your question of right or wrong I'm afraid that the answer is wrong. The phrase that tipped me off was "catch up." Let me explain.
There is a theorem in Statistics called the Weak Law of Large Numbers that states that in independent repeated trials with probability p of success at any given trial, if S(n) represents the number of successes in n trials that for any arbitrary fixed number e, however small, the quantity
Prob[ |S(n)/n - p| > e]
can be made arbitrarily small by making n sufficiently large. In simple terms, this means that it is quite likely that for a large number of trials the ratio S(n)/n is a good approximation to p. In layman terms this is usually called the Law of Averages. In the case of rolling a 9 with a pair of dice it says that S(n)/n is a good approximation to 1/9 for large n.
Notice that although the theorem provides information about the ratio S(n)/n it does not say anything about the actual deviation of S(n) from the average np. Consider the following example in which a fair coin (p = 1/2) is repeatedly tossed. S(n) will measure the number of heads recorded in n trials.
S(n)
S(n)/n
p – S(n)/n
np –S(n)
Notice that the ratio S(n)/n is indeed approaching the theoretical figure 0.5 as n gets larger and larger. In fact, if there were 64,000,000 trials and 31,992,000 heads were recorded, the ratio S(n)/n would be 0.499875; a difference of only 0.000125 from the theoretical. On the other hand notice that the difference in the actual number of heads from the expected number is constantly growing. In the final example it would have grown to 8000. If you were betting $1 on heads at each trial with an even payoff you would be $8000 in the red but the Law of Averages would be right on target!
Now you may think that the above example is atypical. After all, isn't the most likely situation one in which the number of heads and tails are roughly equal? The answer, surprisingly, is no. It can be shown that the most likely situation is that the number of heads either stays above the mean or below the mean for the entire number of trials. The mathematics involved in this is beyond the scope of this article but if you care to investigate this assertion the buzz phrase to look for is The Arcsine Law. In fact it is more likely that the sequence of trials never crosses the mean than it crosses it once, it is more likely that it crosses the mean once than it crosses it twice, and so on. In terms of gambling it means that if you get far behind, though you might catch up, it is likely that you'll never catch up. I'll leave you on that cheery note and see you next month.