Picketing, Pocketing

Suppose each of your pockets can be picked with an equal probability on any given day. I’m going to take a conservative estimate of 10% and assume that the probabilities are independently and identically distributed. You have some money, say Rs. 500 that you would like to carry. Would you prefer to distribute them across both pockets so that the probability of you having nothing is minimised, or would you prefer to keep it in a single pocket to maximise the probability of getting to keep all your money? To put it another way, by putting your money in a lot of different pockets, are you actually raising the probability of at least something getting stolen?

Is it correct to say that the probability of each pocket being picked is independent? We could argue in favour of this assumption because it is unlikely that pickpockets would take the risk of checking all the pockets of the same person once they’ve found something. But if the assumption is correct, then you should be indifferent between the distributions (250, 250) and (499, 1), which you are not. So let us further simplify by assuming that the money is equally divided between all pockets.

Let’s calculate what you should do, shall we? If you put all your money in one pocket, the expected value of the money that you can expect to have at the end of the day is 0.9 * 500 = Rs. 450. Since we assume that the probability of the first pocket being picked is completely unrelated to the probability of the second pocket being picked, the intersection of the events should simply be the product of the two probabilities. So the probability both pockets being picked is 0.1 * 0.1 = 0.01. If you equally distribute your money over both pockets, the expected value will be 500 – P(AUB) = 0.1*250 + 0.1+250 – 0.01*500 = Rs. 455. While the amount you put in each pocket doesn’t affect the expected value, the number of pockets will. So it’s rather rational to keep your money in many different pockets if you’re an average person.

But there could be pickpockets who grab opportunities with both hands. Assuming a probability distribution over all pockets instead of conditioning it on the individual does not allow for a fantastically unlucky individual who has all pockets picked on a single metro ride. Let’s just say nobody likes that guy too much anyway so we aren’t relaxing any assumptions for him. This also means that we don’t think a person’s pocket is more likely to be picked just because he has a lot of them. I’m forced to concede that this, too, isn’t very unrealistic. Where will the pickpocket look in a pair of Dockers?

So even without accounting for risk aversion, our model shows that it is indeed rational not to put all your eggs in one basket. In any case, the preference for not losing everything is probably stronger than the preference for not losing anything. But extend this concept to too many pockets and regardless of what the calculations tell you, you should know that you will end up losing at least some money just by forgetting about it (true story). It is reasonable to say that there is an upper bound to the number of pockets you will split your money across: just as it would be embarrassing to find out that you are broke because your pocket has been picked, it would be rather awkward to have to fish out tenners from four different pockets to pay for a doughnut. The answer, my friend, is moderation.