Friday, February 10, 2012

So far they WON all matches,So will they LOSE the next ? ( The Gambler's Fallacy)

Well they won all the test matches against India. And so far all the matches of the CB series. They are not a very strong team as the previous Australian teams who controlled the world cricket arena for more than dozen years. But recently they have managed to win a fair amount of matches continuously.  Although Australia have the home advantage I think all the three teams have a somewhat fair chance of winning a match. The probability of an Australian win may be high but the probability of two consecutive wins will be low. And of course the probability of three continuous wins will be much lower. So Does this mean that Australia have a very high chance of loosing the next game ? or India have a high chance of winning the match on 12th ? or more importantly if Australia win on 12th Sri Lanka have very very high chance of winning the next match against Australia ?

This leads us to a very interesting fallacy in probability named as the Gambler's fallacy or Monte Carlo fallacy. So its time to think about this fallacy leaving cricket for a moment.

The Gambler's Fallacy

                 The basic idea of the gambler's fallacy is based on the perception we have that a result which we have obtained for a lengthy sequence will change in the next event. If we got 5 or more heads in a row while tossing an unbiased coin most of the time we will expect a tail in the next event. Not only we think but also we may incorrectly derive that the probability of a tail is much higher than 0.5 and reaching to 1. Lets check this using an example. Lets assume we have a fair coin so that the probability of a head or a tail equals 0.5. Lets consider the case where we get 5 consecutive heads. Then the probability will be 0.5^5=0.03125 which is a very rare event. So now consider a situation where we have already got 4 consecutive heads. So the big question is what is the probability of getting another head in the next tossing of the coin ? One may solve the problem in the following way.
Since the probability of getting 5 consecutive heads is 0.03125 the probability of not getting a head is 1-0.03125=0.96875. So the probabilities of getting a head and getting a tail is respectively 0.03125 and 0.96875. This reasoning highly favors a tail in the next tossing of the coin and this leads to the famous Gambler's Fallacy.
                            The problem of the above reasoning is that it does not take the independence of each tossing. It may be true that we had 4 consecutive heads so far. But it does not alter the probability of the next event. What has happened so far is simply the past and unlike in some real life scenarios they don't affect the future. There will be no demons or angels from the past events affecting the future when the events are simply independent.
                            So the probability of getting a head in the next tossing is still 0.5.  This confusion is the basic root of the gambler's fallacy. When a same result is observed for a large number of times people tend to think that the opposite will occur in the future. Based on that assumption they will bet on those results. This simple fallacy will lead to many interesting explanations of some events. Also there may be some instances where the gambler's fallacy may not be applied. As an example assumption of independence of events may not be totally correct. 
                             In the scenario at the beginning the events are not independent. There is a very slight dependence. Due to the consecutive wins they  may decide to give an opportunity to a new player or to rest some senior players. Then the Gambler's fallacy will not become a fallacy any more and the other two teams may increase their chance of winning.