This one is for fans of ABC television entertainment shows and statistics…
It’s a story about the number of ways we can arrange the elements which make up a set, which is called the factorial. But lets start at the beginning.
The new show on the ABC by the team from The Chaser is called Media Circus. In the show two teams compete in various games to answer questions about the news. One of the games is called That’s All We Have Time For. In it each team is given a set of four news items and a set of four time durations. The aim of the game is to match each news item with the time spent on it by the Australian media.
Likewise, the great music quiz show Spicks and Specs which ran on the ABC from 2005 to 2011 had a game called Sir Mix’n’Matchalot in which contestants were given a set of three musicians’ names and a set of three facts with the aim being to match the right musician to the right facts.
In both games, one point is given for each correct match.
The key thing to realise at this point is that, mathematically, these games use the same basic idea but change one important feature: the number of elements in the sets (three in Spicks and Specs and four in The Chaser’s Media Circus).
So what is the basic idea at the heart of both games? The basic idea is to arrange the set given to the contestant (news items or musicians’ names) in the order which matches the answers (time durations or facts).
Sometimes it is easier to see the statistics of a situation if we remove all the real world stuff and replace it with numbers. So lets invent a new game called The Sequence. I give you the numbers one to three and you have to arrange them in the order I am thinking of.
Obviously there is no way to solve this except through chance. So, what is the chance of solving it? If you take your time you can set up a table like the one below with all the possible orders of the set of numbers from one to three.
Possible answers to The Sequence
So there are six ways to arrange a set of three elements. This means that the chance of getting the right answer in The Sequence (or in Sir Mix’n’Matchalot) is 1 in 6.
And what if we changed The Sequence to have four elements like That’s All We Have Time For? For sets larger than three elements this can be time consuming to do by hand. So, we can use something called the factorial. The factorial of a set with n elements can be found by multiplying all the positive integers less than or equal to n. So, for the original Sequence game that would be 3*2*1 which is 6 ( which matches what we did by hand). For the new Sequence that would be 4*3*2*1 which is 120. Which means that the chance of getting the right answer in That’s All We Have Time For by chance alone is 1/120.
So, the first thing you can see if that it is harder to get maximum points in the game on The Chaser’s Media Circus than on Spicks and Specks. But that’s not why it’s a better game statistically.
Its not just that That’s All We Have Time For has more elements than Sir Mix’n’Matchalot, but rather, that That’s All We Have Time has an even number of elements.
This is about the way the games are scored. Remember, players are given one point for each correct answer.
To start thinking about this its useful to think about why we give different points to players in games when different feats are achieved. A goal in the AFL is given six points, a behind one point. In this case I think it is safe to conclude that we give more points to goals because they are harder feats to achieve.
Hence, it would make sense if, on Sicks and Specks and The Chaser’s Media Circus, one point was given for each correct answer to the game because it is harder to get more matches.
Actually, no. That’s not right.
Or rather, that is right for The Chaser’s Media Circus but not for Spicks and Specks.
Below are two graphs which show respectively the number of chances and the probability of getting each number of points available for games such as That’s All We Have Time For and Sir Mix’n’Matchalot with sets which range in size from 1 to 5.
The first thing you might notice is that for each set, there is no way of getting n-1 points. So, in Sicks and Specks, no one ever scored 2 points and in The Chaser’s Media Circus no one will ever score 3 points. This is related to another topic, the degrees of freedom which we can talk about another time.
The bigger problem is with the probability of scoring 1 point. For the game to make any sense, it should be harder to score 1 point than 0 points. This is indeed the case for sets with four elements such as in The Chaser’s Media Circus. In this case the probability of scoring 1 point is 33% and the probability of scoring 0 points is 37.5%. But as you can see, for sets with three elements (such as in Spicks and Specks), the probability of scoring 1 point by chance alone is 50%, but the probability of scoring 0 points is actually less, at 33%. This is the equivalent of AFL players scoring 6 points for a goal, 1 point for a behind but 10 points for missing everything. The reward is greater for the easier task.
If you look at the other lines you might see a pattern, for sets with 1, 3 and 5 elements, it is harder to get one point than 0 points. For sets with 2 and 4 elements it is easier to score 1 point than no points at all. This is shown again for a larger range of set sizes in the figures below.
To understand this it might help to remember what the points are gained for: getting one of the matches right. So, for the scoring system to make sense, there should be more ways of arranging the elements of the set so that none of them match the answer than ways in which one of them matches the answer.
And here is the difference between sets with odd and even numbers of elements: one of the ways we can arrange the elements is the reverse of the correct answer. For sets with an even number of elements, this arrangement scores no points because none of the elements match the answer. But as you can see in the table below, for sets with an odd number of elements, this arrangement gains one point because the middle element is correct.
Three member set
Four member set
Hence, the new show by the team from The Chaser has a scoring system which more appropriately rewards players for achieving feats of greater difficulties. Unlike Spicks and Specs, which got things a bit back to front.