Monte Carlo Example: Pólya’s Lucky Dip

Edit (2013/10/04): Under the first picture below, I mention a line which should be on all the pictures, but isn’t. This line should be at around 25 for all of them.

Probability has a few standard analogies. Let’s get to grips with one of them.

Let’s say we sit Ronald down in front of a bucket of red and blue magnetic toy fish, and the red fish have a prize written on them. He then catches one of the fish with a magnetic rod. It turns out to be blue, so he stores the fish by clipping it to his beard, and tries again. Assume the rather unlikely case where he knows that there are ten red fish and a hundred blue ones. What’s the chance of winning if he can catch up to three fish?

The standard mathematical way to solve this is to compare the picking-out of fish to one of the classic examples of picking balls out of an urn. But, frankly, it’s early in the morning, and I don’t want to deal with binomial coefficients before I’ve had a few drinks. What I could do with is convincing Ronald to sit around playing lucky dips all day, and see how often he wins. Since it’s a bit late to ask Ronald to fish, I’ll use a computer instead.

If we’d had Ronald to do this, I’d have to choose how many times he got to have a go. Since we’re using a computer, I’m just going to pick a few different numbers, and see what difference it makes. After a break for a drink, this is what I came back to.

Fig1

The dots are the guesses, the line is what I know the true probability to be. However, if I run this again, the result can be very different.

Fig2

Our guesses have a random element to them, so that shouldn’t be surprising. If I let Ronald play ten times, and then ten times again, the two sets of results needn’t have the same number of winners. What this means is that, if I make several guesses with the same number of plays, they’re going to be spread out. Since, in practice, we’re only going to make one guess, we’d like the spread to be pretty small. Hopefully, we can achieve this by increasing the number of plays.

I reworked the program to take a hundred guesses for each number of plays, and then use them to draw boxplots. After a bigger drink, I came back to this.

Fig3

If you’re not used to boxplots, half of the estimates are inside the box, and the other half are inside the dotted lines. Dots are outliers that I’m just going to ignore here. The boxes get smaller as the number of entrants increases, and the box for 10000 plays is tiny. In other words, increasing the number of plays decreases how spread out the estimates will be.

What we can say is that, if you wanted your guess to be precise to within a percentage point or two, you’d need to simulate about ten thousand goes. Maybe it’s just as well we didn’t ask Ronald, I’m out of drinks to bribe him with for that long.

Does the box shrink towards the correct value? We don’t know, unless we work it out. In this case, my throat is now wet enough that I feel up to working it out on paper. In this case the true probability of winning is \frac{82}{327}, or about 25%, so it looks like the guesses tend towards the correct value as we increase the number of plays. It also looks like this would be a terrible lucky dip from the point of view of the person paying for the prizes, but never mind.

This is mainly because we know how to make direct guesses. By that, I mean the new data we’re generating is a direct statement about what we think the probability is. What we didn’t have to do, for example, was to be given two sets of Ronald’s win frequencies, and have to guess at whether he was fishing from the same bucket each time. We could generate more win frequencies, but those aren’t something we can directly use as a statement of whether or not the bucket is the same.

This requires more clever methods, and these more clever methods don’t necessarily tend to the correct answer if we run them for longer. That might be because the method we decide to use is very good. It might be because the data we can generate is so uninformative about the answer, that deriving one is going to introduce errors, regardless of what we do. But that deserves a separate post.

Why didn’t I just have a few drinks first, and go straight to getting the right answer? Well, I could have done, but sometimes you don’t have that option. More on that another time.

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