Although this is based on a book/film I think the question itself is more GQ.
Just watched Hunger Games: Catching fire and wondering at the mathematics of the reaping. Minor spoilers for the film follow.
So, there have been 74 Hunger Games, and in that period there is at least 1 male and 1 female from all 12 districts, all 24 possible winners have happened once. That seems unlikely to me even if everybody had an equal chance, but they stated that the richer districts won much more often.
So could somebody do the maths for me?
i)In 74 attempts with 24 possibilities, what is the chance that every possibility happens at least once, assuming each possibility is equally likely?
ii)Judging by the ages of the characters, none of them seemed to have won more than 40 years ago, (except Mags, but she was volunteering for a much younger woman so the younger one could have been there instead). What are the odds of all 24 possibilities coming up in a 40 year period?
i) Let’s assume that the other 23 possibilities are the only ones that ever came up - in other words, there is 1 possibility that NEVER came up. The probability of this is (23/24)^74, which is approximately 4.29%.
Assuming that TWO possibilites never came up - the probability of that is 0.16%.
The chance that there are three or more possibilities that never come up is exceedingly small.
So, to find the probability that ALL the possibilities happen at least once, we can do 100% - 4.29% - 0.16% - a bunch of exceedingly small numbers = 95.55%. So it’s pretty likely that every possibility will have come up.
ii)
We can repeat this with a 40-year period.
Probability 1 possibility never came up = (23/24)^40 = 18.22%.
Probability 2 never came up = (22/24)^40 = 3.08%
Probability 3 never came up = (21/24)^40 = 0.48%
Then things get exceedingly small again.
Probability that all the possibilities happen in a 40-year period = 100 - 18.22 - 3.08 - 0.48 = 78.22%
I suspect there’s a quicker, more accurate, more direct way, but for a non-life-threatening question, this should do.
That doesn’t work. That 4.29% is the probability a specifically named position (e.g the female from district 12) doesn’t ever win, not the general probability of there being exactly 1 non-winner. All the rest of your post was extrapolated from this faulty assumption.
Your math is for any given district never winning. But that’s not what’s being asked. We want the chance of any district never winning. To a good approximation, this is equal to the sum of the chances of each district not winning minus the probability of two districts not winning, i.e. 244.29%-.52423.16% = 59%. Which would put the probability of all of them having won it at 41% (I’ve also ignored the higher order terms, i.e. having three of them not winning. Is this valid to do?)
This seems like it would be easy to simulate; I always confuse myself with probability.
OK, I did a quick simulation, with ten thousand 74 year periods. In a total of 3187 of them did each district win at least once, which would suggest about a 31.9% chance of each district winning once. This is a binomial distribution, so we’re looking at something like 31.9 +/- 0.9% at the 95 percent confidence interval.
Where did that come from? I have scanned the book, but the only reference I could find was that District 12 has had ‘exactly 2 winners’ and only one is still alive.
I haven’t read the books, but in the 2nd film “Catching Fire” they have a special tournament consisting only of past victors. And they fill up every place.
Looking at this page, five of the contestants won over 40 years ago. Mags, Woof, the two drug addicts who were into camouflage and Seeder all won 40+ years prior to the movie. You could still disqualify Mags for statistical purposes since there were 3 known living winners from her district.
If I did the computation correctly (messing around with generating functions in Mathematica— the result was exact before I converted it into numerical form), the probability is p = 32.2036% for n = 74 trials and p = 0.181% for n = 40 trials. I can’t reproduce the details here, but the results agree with Pasta’s Monte Carlo runs.
Not exactly relevant to the OP, but I’d like to add that the implication from the books is that the requirement that the contestants in the Quarter Quell be past winners was created on the fly to ensure that Katniss had to play again.
So if the math hadn’t worked out, they would have had something different for the QQ.
This is one of my peeves about the novel. I just never thought that the odds of there being at least one male and one female winner from each and every district added up, especially the way the harped on the fact that the career districts usually won with only an occasional winner from the outlying districts,
For those doing the math at home, the book does state that at the time of the Quarter Quell there were 59 surviving winners
I got more the impression that the Careers won a disproportionate amount of the time, but that there were more than “occasional” winners from other districts. Katniss talks about how the years the Careers don’t protect their food adequately are the years other districts usually win, and she says “like as not” one of them will win (the phrase stuck out to me because it’s the closest she ever comes to any sort of regionalism that marks her as being from Appalachia), and that generally means about half the time, give or take. For 25% of the pool to win even 40% of the time is a pretty significant advantage, and would mean that any given non-Career district does indeed win only occasionally.
If the Careers win a full 50% of the time, that leaves 37 remaining Games, and 18 remaining spots to fill. The odds of every single spot hitting at least once aren’t great, but such a thing happening is at least as likely as invisible hovercraft.