Suppose I want to estimate my chances of winning the lottery by buying a ticket every day. That is, I want to do a pure Monte Carlo estimate of my probability of winning. How long will it take before I have an estimate that’s within 10% of the true value?

**It’ll take…**

There’s a big NY state lottery for which there is a 1 in 300M chance of winning the jackpot. Back of the envelope, to get an estimate within 10% of the true value of 1/300M will take many millions of years.

**Didn’t you say Monte Carlo only took a hundred draws?**

What’s going on? The draws are independent with finite mean and variance, so we have a central limit theorem. The advice around these parts has been that we can get by with tens or hundreds of Monte Carlo draws. With a hundred draws, the standard error on our estimate is one tenth of standard deviation of the variable whose expectation is being estimated.

With the lottery, if you run a few hundred draws, your estimate is almost certainly going to be exactly zero. Did we break the CLT? Nope. Zero has the right absolute error properties. It’s within 1/300M of the true answer after all! But it has terrible relative error probabilities; it’s relative error after a lifetime of playing the lottery is basically infinity.

The moral of the story is that error bounds on Monte Carlo estimates of expectations are absolute, not relative.

**The math**

The draws are Bernoulli with a p chance of success, so the standard error of the Monte Carlo estimator

is going to be its variance

for draws and a

probability of winning.

**Extra credit: Sequential decision theory**

How long would it take to convince yourself that playing the lottery has an expected negative return if tickets cost $1, there’s a 1/300M chance of winning, and the payout is $100M?

Although no slot machines are involved, this is the simplest framing of a so-called “bandit” problem. More sophisticated problems would involve several lotteries with generalized payout schedules that might be stateful or non-stationary.