mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

Pointless probability

 2013-12-15 
Last week, I was watching Pointless and began wondering how likely it is that a show features four new teams.
On the show, teams are given two chances to get to the final—if they are knocked out before the final round on their first appearance, then they return the following episode. In all the following, I assumed that there was an equal chance of all teams winning.
If there are four new teams on a episode, then one of these will win and not return and the other three will return. Therefore the next episode will have one new team (with probability 1). If there are three new teams on an episode: one of the new teams could win, meaning two teams return and two new teams on the next episode (with probability 3/4); or the returning team could win, meaning that there would only one new team on the next episode. These probabilities, and those for other numbers of teams are shown in the table below:
 No of new teams today
Noof new teams tomorrow
  1234
100\(\frac{1}{4}\)1
20\(\frac{1}{2}\)\(\frac{3}{4}\)0
3\(\frac{3}{4}\)\(\frac{1}{2}\)00
4\(\frac{1}{4}\)000
Call the probability of an episode having one, two, three or four new teams \(P_1\), \(P_2\), \(P_3\) and \(P_4\) respectively. After a few episodes, the following must be satisfied:
$$P_1=\frac{1}{4}P_3+P_4$$ $$P_2=\frac{1}{2}P_2+\frac{3}{4}P_3$$ $$P_3=\frac{3}{4}P_3+\frac{1}{2}P_4$$ $$P_4=\frac{1}{4}P_1$$
And the total probability must be one:
$$P_1+P_2+P_3+P_4=1$$
These simultaneous equations can be solved to find that:
$$P_1=\frac{4}{35}$$ $$P_2=\frac{18}{35}$$ $$P_3=\frac{12}{35}$$ $$P_4=\frac{1}{35}$$
So the probability that all the teams on an episode of Pointless are new is one in 35, meaning that once in every 35 episodes we should expect to see all new teams.
Edit: This blog answered the same question in a slightly different way before I got here.

Similar posts

Countdown probability, pt. 2
Countdown probability
Big Internet Math-Off stickers 2019
World Cup stickers 2018, pt. 3

Comments

Comments in green were written by me. Comments in blue were not written by me.
 Add a Comment 


I will only use your email address to reply to your comment (if a reply is needed).

Allowed HTML tags: <br> <a> <small> <b> <i> <s> <sup> <sub> <u> <spoiler> <ul> <ol> <li>
To prove you are not a spam bot, please type "rotcaf" backwards in the box below (case sensitive):

Archive

Show me a random blog post
 2020 

Mar 2020

Log-scaled axes

Feb 2020

PhD thesis, chapter ∞
PhD thesis, chapter 5
PhD thesis, chapter 4
PhD thesis, chapter 3
Inverting a matrix
PhD thesis, chapter 2

Jan 2020

PhD thesis, chapter 1
Gaussian elimination
Matrix multiplication
Christmas (2019) is over
 2019 
▼ show ▼
 2018 
▼ show ▼
 2017 
▼ show ▼
 2016 
▼ show ▼
 2015 
▼ show ▼
 2014 
▼ show ▼
 2013 
▼ show ▼
 2012 
▼ show ▼

Tags

sound world cup graphs football cross stitch pac-man gaussian elimination game of life bodmas captain scarlet game show probability triangles convergence weak imposition javascript hats harriss spiral plastic ratio gerry anderson speed preconditioning polynomials curvature computational complexity propositional calculus final fantasy countdown data news inverse matrices martin gardner php signorini conditions mathsjam rhombicuboctahedron books stickers reddit flexagons dates estimation london underground sobolev spaces platonic solids twitter chalkdust magazine machine learning inline code wool probability data visualisation geometry coins phd graph theory european cup a gamut of games rugby advent calendar golden ratio matt parker fractals ternary matrix multiplication talking maths in public python simultaneous equations electromagnetic field cambridge light london error bars frobel determinants weather station games braiding golden spiral go mathsteroids sorting accuracy logic latex reuleaux polygons bempp nine men's morris christmas card sport interpolation matrices national lottery christmas numerical analysis big internet math-off the aperiodical craft trigonometry oeis chebyshev royal baby manchester hannah fry statistics chess menace mathslogicbot ucl logs people maths raspberry pi puzzles misleading statistics bubble bobble matrix of cofactors pythagoras palindromes folding paper radio 4 dataset approximation wave scattering boundary element methods realhats draughts folding tube maps royal institution binary tennis arithmetic map projections dragon curves finite element method pizza cutting noughts and crosses matrix of minors hexapawn exponential growth tmip video games manchester science festival asteroids programming

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2020