Luk Arbuckle

Debunking the “law” of averages

In probability on 12 June 2008 at 2:40 am

One example of the “law of averages”—not to be confused with the law of large numbers—describes the belief that a particular event becomes more or less likely to occur in order to move a series of random events closer to the long-run average.  A coin toss, for example, is believed more likely to come up tails if ten heads have been successively thrown beforehand.  The belief is that the long-run average of 50/50 must be made up for, and therefore the probability of the single event must change to suit that long-run average.

It’s not surprising that anyone would believe the law of averages, even if it is misguided.  Although the coin isn’t aware of what was thrown beforehand, there is the notion of a long-run average to contend with.  The binomial distribution, used to consider success and failure experiments like coin tosses, can be used to show why this law is false.  But we’ll skip the math and only discuss results.

Watch your heads
The variability of the number of successes increases with an increasing number of trials.  You toss a coin 10 times and get 10 heads: 5 heads more than average.  You toss a coin 100 times and get 60 heads: 10 heads more than average.  That’s an increase in the variability of the number of successes from 5 to 10.  But clearly you’re getting closer to the 50/50 split of heads and tails.  The problem, then, is that we’re considering the wrong measure.

The variability of the proportion of successes, however, decreases with an increasing number of trials.  You toss a coin 10 times and get 10 heads: a proportion of 10 over 10, or 1.  You toss a coin 100 times and get 60 heads: a proportion of 60 over 100, or 0.6.  As we toss the coin more times, the proportion of heads (heads divided by the number of tosses) gets closer to the 50/50 average we’re expecting.

Throwing ten heads in succession will not change the probability of getting heads in the eleventh coin toss: it’s still 50/50 heads or tails.  But as the coin is repeatedly thrown those ten heads will have less effect on the proportion of heads that come up.