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 "raenil" backwards in the box below (case sensitive):

Archive

Show me a random blog post
 2020 

May 2020

A surprising fact about quadrilaterals
Interesting tautologies

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

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

Archive

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