Gamblin' man

It’s no secret to those who know me (and now you) that I enjoy gambling. Why so many of us engage in this insidious behaviour has been a focus of psychology research for many years. Personally I like to think that my tendency to gamble arises from an interest in and basic understanding of statistics. You only need to be vaguely familiar with the concept of gambling or perhaps have access to a dictionary to argue that I’m a fool. In my defence, I’m certainly not the type who strolls into a casino thinking I can win a fortune. Casinos and betting agencies employ statisticians far more competent than me to ensure that they win and I lose.

So when to apply the statistical knowledge endowed to me by psychology for my own personal gain? Fortunately people in general are far less adept at identifying true statistical patterns and reasoning out statistical problems than casinos. Last week I was in a bar with a friend who remarked “I’m not sure I know anyone who shares my birthday”, immediately I saw an opportunity to make some money. I bet them that there would be two individuals in the bar who shared a birthday. Was this a good bet?

My friend and I estimated there were at least 30 people in the bar, and being offered 2:1 odds that I was correct, I jumped at the bet. Did I make the correct decision? To answer this you need to know if the return on my bet is better than the odds of winning. That is, how many people are required in a group for there to be a greater than 50% chance that two of them share a birthday? If the number of people required exceeds 30 then I have made a bad bet.

The problem becomes easier to solve if you work out the probability that no one will share a birthday. In a group of two people the chance of no match is 364/365. Add a third person and the chance they match the first is 1/365 and the chance they match the second is also 1/365. Therefore the chance the third person does not match either the first or second is 363/365. To calculate the odds we multiply the fractions, 364/365 x 363/365, which gives a 99% chance that no one in a group of three shares a birthday or a 1% chance that they do.

As each person is added, the odds do not increase linearly, but rather they curve upwards rapidly. If you continue to apply this logic for a group of 4, 5, 6 and so on, up until a group of 23 people, you find that there is a 51% chance two people will share a birthday. It turns out in a group of 30 people there is a 70.1% chance that two people with share a birthday.

Without out the aid of formalised statistical methods it’s difficult in this case to arrive at the correct decision. This is why I enjoy statistics, they can often show the seemingly improbable to be quite the opposite. However, I didn’t actually compute these odds at the bar; I knew of them prior to the bet and thus was essentially hustling my good friend...



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