I recently finished reading the first two books of the three-book series *The Hunger Games*, by Suzanne Collins. The series, and the movie currently in theaters, seem to be extremely popular with a lot of teens right now. Last week I asked a group of students if they had read the books; every one of them had read at least the first book, and most of them had read all three.

Lately, the story seems to be popular with math teachers, too. A frequently referenced recent *Wired* post comments on the probability and game theory involved in the lottery-like selection of “tributes” and their strategy during battle in the arena. When students are already interested in the story, it can be a great opportunity to motivate discussion of related interesting mathematics.

Frankly, I do not think these are great books. Good, yes, with an interesting and thought-provoking premise. But at times they feel like fast-food writing, with quite a few gods in the machine to keep the plot stitched together. In a 2008 review, Stephen King points out some examples in the first book of what he calls “authorial laziness.” The motivation for this post is one particular episode in the second book, *Catching Fire*, that arguably also fits that bill… but which can also be viewed more optimistically as an opportunity to discuss the non-intuitive nature of probability with students.

**Spoiler Alert: The following may give away important plot points in Catching Fire. You have been warned.**

The story takes place in post-apocalyptic North America, where the country has been divided into a controlling Capitol and 12 Districts. As punishment for a long-ago rebellion, every year each district must send one boy and one girl, called tributes, to take part in the Hunger Games, where the 24 tributes fight each other to the death in an arena until one victor remains. That victor is rewarded with a life of wealth, luxury… and freedom from participation in future Games.

However, on the 75th anniversary of the rebellion, the evil Capitol decides to add a twist to that year’s Hunger Games: instead of selecting tributes from their young boys and girls, each of the 12 districts must instead send one male and one female from its pool of *past victors*, some of which might now be old men or women. As described in the book:

In the history of the Games, there have been seventy-five victors. Fifty-nine are still alive… As one would expect, the pools of Career tributes from Districts 1, 2, and 4 are the largest. But every district has managed to scrape up at least one female and one male victor.

Let’s re-state the setup, just to be clear: among the 59 living victors of past Hunger Games, each of which could be male or female, and from any of the 12 Districts, there is fortunately at least one female and one male from each district, to make up the 24 contestants for the 75th Games.

ObPuzzle: Making any necessary assumptions, how unlikely is this scenario? That is, what is the probability that, among 59 victors emerging from the male and female populations of 12 Districts, at least one male and at least one female from each district is represented?

I like this problem for several reasons. First, at first glance it may not *look* like a problem at all. When considering situations involving some element of randomness, our intuition is notoriously unreliable. This particular example highlights our tendency to equate “random” with “evenly distributed.” Second, the problem may be attacked in several different ways: computer simulation, exact analysis, etc.

As usual, solutions/discussion are welcome in the comments.

If we disregard the tidbit of “As one would expect, the pools of Career tributes from Districts 1, 2, and 4 are the largest” and assume equal probability for each district and for each gender, then the problem becomes solvable.

I wish I remembered more probability theory, because solving this analytically seems very possible and interesting, but I tackled it via computer simulation. I first thought of the problem a different way: Imagine you have 24 buckets, and a Price Is Right “Plinko” device (where you drop a disc and it traverses down a series of pins and obstacles, eventually falling into one of the buckets). You also have 59 discs. Let’s also assume that the probability a disc will drop into any given bucket is the same (uniform distribution) and is also independent of the previous drop. Now, to me, the problem seems easier to translate into a computer simulation. What we are really doing is picking a random number between 1 and 24 (inclusive) 59 times. After the 59 times, if we have picked each number at least once, then we have all our “victors” for the hunger games. Think of each number as a specific gender from a district. Number 1 represents a male from district 1, number 2 represents a female from district 1. Number 3 is a male from district 2, and 4 is a female from district 2. So on and so on until 23 is a male from district 12 and 24 is a female from district 12.

After I ran my simulation for 2 million iterations, I got slightly less than 10% of the time (9.92%). So how unlikely is the scenario given in the book? 9 times out of 10, it doesn’t happen, so I’d say that is pretty unlikely.

This is also assuming that I didn’t make a mistake in my code anywhere. I was interested in what the percentage would be for different numbers (if we only needed 14 victors (i.e. there are only 7 districts) or if we had the whole 75 person pool instead of 59). I ran lots of these simulations, and what was interesting was having 48 person pool for 12 districts only gave you a 1.75% chance of having a victors from each category, 75 people for 12 districts gave you a 34% chance, and you needed a 345 person pool for it to be “certain” (meaning C++’s standard double data type rounded up to 100% when it did its division).

I like the comparison with Plinko :). Your simulation results are right on; I will follow up shortly with another post describing the analytic solution.

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heh. i like this problem enough to go looking to see if anyone else had tackled it.

um, for the 75th games aren’t there 74 victors to reap from? or am i confused?

in the books you know 3 of the victors are from district 12. so another question is: given 56 victors of unknown gender, what’s the probability districts 1-11 all have at least one male and at least one female victor?

You’re not confused; I suspect this was simply a mistake on Collins’ part, but I’m not sure.

The follow-up post describes the general solution to this problem. To answer your question, plug in the values 56 for n and 22 for d; the resulting probability is about 0.15.

There were actually 75 victors because the first 73 had one victor each, and the 74th Hunger Games actually had two winners.

Ah, you’re right. I forgot that the previous games (the main focus of the first book) had two winners. Thanks!