On the Wizard of Odds’ Website there’s a “Gambler’s Fallacy FAQ” where there’s an exchange between the Wizard and a guy who keeps asserting the Gambler’s Fallacy, i.e. that because there is a cumulative-probability calculation that says the probability of n events(x) in a row is (a small) p, that after n-1 events(x) the probability of another (x) is very, very small and the probability of non-X is very high.

The basis for this argument is that the probability of the series of events does not change as the events in the series take place, rather than the probability of each event remaining constant. There’s a very easy way to explode this argument. Suppose we flip a fair coin three times. Here are the possible outcomes:

HHH
HTH
HTT
HHT
THH
THT
TTH
TTT

There are only eight possible arrangements of heads and tails for three throws, and they are equally likely. So the probability of each is .125. By definition, impossibility is zero and certainty is one, with all other outcomes somewhere between.

So, before we throw a coin, the probability of HHH is .125. The Fallacy would argue, then, that after two heads have been thrown, the probability of the next flip being a head is just .125, while the probability of a tail must be .875, since that is the only other possibility. IOW, the individual-event probabilities change, while the cumulative probabilities stay the same.

Here, again, are the eight possibilities and their associated probabilities:

Now, we can combine some of these if we ignore the order and just concentrate on the number of each outcome:

So, let’s step through the series and examine what happens, keeping in mind that the total of the probabilities of possible outcomes must always equal 1.000.

We flip the coin! Result, heads!

HHH
HTH
HTT
HHT
X THH
X THT
X TTH
X TTT

Now, one of the three events is now determined; since the first flip was heads, all of the originally-possible outcomes starting with a tail are now eliminated, right? If we started out with a .125 probability of three tails, what is the probability of TTT now? It’s zero, isn’t it? In fact, all four of the possible outcomes starting with a tail are now impossible, with probabilities of zero, right? Now, since the total of the probabilities of the possible outcomes must always equal 1.0, what happened to the 4 * .125 for those four starting-with-a-tail outcomes? THEY HAVE TO BE RE-DISTRIBUTED among the still-possible outcomes. We are left with:

HHH
HTH
HTT
HHT
which are all equally likely, so each has a probability of .250, summing to 1.000.

So, we see very clearly and unambiguously that the probabilities associated with a series of events MUST CHANGE as the events unfold. TTT started out at .125, but it is now 0.000.

This is the end of the argument for the Gambler’s Fallacy. It clearly fails. In fact, we only need to consider two events:

HH .250
HT .250
TH .250
TT .250

First flip is heads. Now TT and TH can no longer happen, right? End of story. Game over. p(TT) and p(TH) are now 0.00 and HH and HT are each .5.

It’s really such a simple concept, and yet some quite intelligent people persist in believing that, if the probability of six sevens in a row is just .000021433, then after five sevens in a row the probability of another seven is very, very small. Or, they believe that, if the probability of six dice rolls without a seven is just .3349 (.8333333^6), that after five rolls without a seven, the probability of the next roll being a seven is .6651, not .1667.

The odd thing is that all these calculations of cumulative probability are based on a single probability for the event in question. Duh! So these people recognize a constant probability of a single event, then turn around and deny it. Well, it’s anti-intuitive, isn’t it?
Cheers,
Alan Shank