The Hunger Games is the 2012 blockbuster first movie in the Hunger Games series, based on the book series by Suzanne Collins. Set in an apocalyptic future, teenager “tributes” from downtrodden districts must compete in the life and death “Hunger Games” that enforces the rule of the ruthless Capitol. There are a number of math-filled scenarios in the movie including archery and lotteries.
You can seen the trailer here:
Math Scenario 1: The Hunger Games Lottery
When kids turn 12 they become eligible for the Hunger Games lottery and continue to be eligible up to and including age 18. At that point in time, their name is entered once in the lottery.
If they are not drawn that year, then the next year when they’re 13, they have two entries in the lottery with their name – increasing their chances of being picked.
This continues until they are 18, when presumably they have seven entries in the lottery.
Large Population of Kids with Evenly Distributed Ages
If we assume that the population of children between the ages of 12 to 18 stays approximately constant, is large (in the movie it looks like there are many hundreds of eligible children) and evenly distributed over that age range, then we can work out how much worse off you are as an 18 year old then a 12 year old.
The 12 year old has a single entry in the lottery
The 18 year old has seven entries in the lottery
Therefore the 18 year old is seven times more likely to be chosen than a 12 year old.
Small Population of Kids All Aged 12
If however, you’re unlucky enough to live in a small district with children under the age of 12 and a small number – say 20 children – just turning 12, then things are a little different.
20 children aged 12 each with one entry into the lottery
1/10th chance of being selected (2 children selected from each district)
After year 1, two children have either not made it back from the hunger games alive or won and are now no longer eligible. So there are only 18 children left for consideration.
18 children aged 13 each with two entries into the lottery
1/9th chance of being selected (2 children selected from each district)
16 children aged 14 each with three entries into the lottery
1/8th chance of being selected (2 children selected from each district)
14 children aged 15 each with four entries into the lottery
1/7th chance of being selected (2 children selected from each district)
12 children aged 16 each with five entries into the lottery
1/6th chance of being selected (2 children selected from each district)
10 children aged 17 each with six entries into the lottery
1/5th chance of being selected (2 children selected from each district)
8 children aged 18 each with seven entries into the lottery
1/4th chance of being selected (2 children selected from each district)
This is pretty grim – anyone who survives to age 18 then has a 25% chance of being selected this year.
Individual with More Entries
The third scenario that could occur is that people could nominate their child to be entered more times than normal in exchange for more food or other critical supplies. We can see what effect that might have on a child’s chances of being selected. Let’s examine a child aged 12, who has had an extra 10 entries in exchange for food, bringing their total up to 11 (instead of the normal 1 by that age). Let’s say they’re in a district with 70 other kids (10 aged 12, 10 aged 13…) who have the normal number of tickets for their age. Compare the chances of this special 12 year old to what they would have been if they were just a normal 12 year old with one entry.
Total other entries by other kids = aged 12 entries + age 13 entries + … age 18 entries
Total other entries by other kids = 10 * 1 + 10 * 2 + 10 * 3 + … 10 * 7
Total other entries by other kids = 280
A normal 12 year old would have one entry, so their chances of being selected would be:
Normal 12 year old odds = their entries / total entries
Normal 12 year old odds = 1 / (280 + 1)
Normal 12 year old odds = 1 / 281
Normal 12 year old odds = 0.36%
Now let’s look at the special case 12 year old:
Special case 12 year old odds = their entries / total entries
Special case 12 year old odds = 11 / (280 + 11)
Special case 12 year old odds = 11 / 291
Special case 12 year old odds = 3.78%
Trading more entries for food makes a radical difference in your chances of being selected:
Increased likelihood = new chance / old chance
Increased likelihood = 3.78 / 0.36
Increased likelihood = 10.5 times higher
Math Scenario 2: Archery to Impress
During the scoring scene in the movie, where Katniss is trying to impress the judges, she pulls off a memorably shot where she shoots an apple out of the banquet pig’s mouth.
Forgetting for the moment things like the projectile travelling along a curved trajectory due to gravity, we can examine how still Katniss would have to hold the bow to pull off the shot.
Let’s assume that she’s about 50 metres away from the gathering of judges where the apple is. An average apple is about 8 cm in diameter.
Ignoring things like fall of the arrow in flight, we can work out the angular accuracy she will need to shoot with to hit the apple at 50 m.
tan(hit angle range) = opposite / adjacent
tan(hit angle range) = 8 cm / 50 m
tan(hit angle range) = 0.08 m / 50 m
Now since this is a very small angle, we can use the approximation that tan(angle in radians) ~= angle in radians:
tan(hit angle range) = 0.08 m / 50 m
hit angle range = 0.08 m / 50 m
hit angle range = 0.0016 radians
hit angle range = 0.092 degrees
That means the total angular range in which Katniss would hit the apple is less than a tenth of a degree – that’s pretty ridiculous shooting.
Real-Life Example – The Prisoner’s Dilemma
The Prisoner’s Dilemma  is a famous theoretical game scenario which is relevant to the survival game at the heart of the Hunger Games. The scenario goes like this:
Two criminals are arrested and put in separate cells with no communication.
The prosecutor offers each prisoner a choice: they can betray the other prisoner or remain silent
There are three possible outcomes:
Both prisoners betray each other and they get 2 years in jail
One prisoner betrays the other but the other stays silent: in this case the prisoner who does the betraying gets set free, and the one who remained silent gets 3 years jail
Both prisoner remain silent and they both get given 1 year in jail
No other factors are considered – the prisoners will never see each other again, there’s no long term repercussions beyond the jail sentence and what they do is all kept confidential so no-one else will ever know about it.
In the Hunger Games, some of the tributes band together – but there is always the choice hanging over them of whether to keep trusting their companions, or to betray them while they sleep. This is a similar problem to that encountered in the Prisoner’s Dilemma.
In real-life prisoner tests, humans tend to be more inclined towards cooperating together than would be predicted by a purely rational analysis. Which is perhaps a good thing for humanity!