Thursday, September 8, 2011

The Lottery

I sometimes hear scientists who are not good at math modeling to justify why it's stupid to buy a lottery ticket. They calculate some probabilities, and think "there are X number of tickets, and Y many number combinations", etc and then say it's unlikely for anyone to win, so why bother?

I argue their model is wrong. They are not taking into account cost / benefit, and the probability of payoff properly. Here is an alternative model:

benefit / cost * probability of winning.

In this model, if for example each lottery ticket is one in a million, costs 1 dollar, potential payoff is 500K, calculation is

benefit = 500000.
cost = 1.
prob = 1./1000000.
print (benefit/cost)*prob

But let's not leave it here. Let's calculate another entity to compare this _to_. For example .. a paycheck. For example someone earns 30K a year, and puts in 40K of effort to earn that paycheck (employee has to be putting in more than he gets, right, otherwise his empoyer makes no money), and he gets that paycheck with probability 1. There is no chance he will *not* get his paycheck, a paycheck in that sense is the opposite of a lottery ticket -- it is almost impossible to win the lottery, it is almost impossible not to get a paycheck. That calculation is

benefit = 30000.
cost = 40000.
prob = 1.
print (benefit/cost)*prob

The results are, for lottery 0.5, for paycheck 0.75. Paycheck wins, but just barely. There is no difference in scale here, and numerical difference certainly isn't something to sneer at. If our made up numbers were massaged a little, the lottery ticket could well come close. Say this employee works like a dog and puts in 60K worth of effort. In that case, we have 0.5 for both calculations.

Then the question isnt why are so many people buying a lottery tickets. The question is why isn't everyone?