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 "uncountable" in the box below (case sensitive):

Archive

Show me a random blog post
 2021 

Jan 2021

Christmas (2020) is over
 2020 
▼ show ▼
 2019 
▼ show ▼
 2018 
▼ show ▼
 2017 
▼ show ▼
 2016 
▼ show ▼
 2015 
▼ show ▼
 2014 
▼ show ▼
 2013 
▼ show ▼
 2012 
▼ show ▼

Tags

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

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2021