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 "t" then "h" then "e" then "o" then "r" then "e" then "m" 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

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

Archive

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