Countdown Probability, pt. 2

As well as letters games, the contestants on Countdown also take part in numbers games. Six numbers are chosen from the large numbers (25,50,75,100) and small numbers (1-10, two cards for each number) and a total between 101 and 999 (inclusive) is chosen by CECIL. The contestants then use the six numbers, with multiplication, addition, subtraction and division, to get as close to the target number as possible.

The best way to win the numbers game is to get the target exactly. This got me wondering: is there a combination of numbers which allows you to get every total between 101 and 999? And which combination of large and small numbers should be picked to give the highest chance of being able to get the target?

To work this out, I got my computer to go through every possible combination of numbers, trying every combination of operations. (I had to leave this running overnight as there are a lot of combinations!)

Getting Every Total

There are 2646 combinations of numbers which allow every total to be obtained. These include the following (click to see how each total can be made):

• 5 6 8 9 10 100
• 5 6 7 8 10 100
• 4 6 7 8 9 100
• 3 6 7 8 10 100
• 3 5 7 8 9 100
• 2 5 6 8 9 100
• 2 6 7 8 9 100
• 5 6 8 9 75 100
• 3 6 8 10 75 100
• 2 6 9 10 75 100

By contrast, the following combination only allows 13 totals between 101 and 999 to be reached:

• 1 1 2 2 3 3

The number of attainable targets for each set of numbers can be found here.

Probability of Being Able to Reach the Target

Some combinations of numbers are more likely than others. For example, 1 2 25 50 75 100 is four times as likely as 1 1 25 50 75 100, as (ignoring re-orderings) in the first combination, there are two choices for the 1 tile and 2 tile, but in the second combination there is only one choice for each 1 tile. Different ordering of tiles can be ignored as each combination with the same number of large tiles will have the same number of orderings.

By taking into account the relative probability of each combination, the following probabilities can be found:

Number of large numbers	Probability of being able to reach target
0	0.964463439
1	0.983830962
2	0.993277819
3	0.985770510
4	0.859709475

So, in order to maximise the probability of being able to reach the target, two large numbers should be chosen.

However, as this will mean that your opponent will also be able to reach the target, a better strategy might be to pick no large numbers or four large numbers and get closer to the target than your opponent, especially if you have practised pulling off answers like this.

Edit: Numbers corrected.

Countdown Probability

On Countdown, contestants have to make words from nine letters. The contestants take turns to choose how many vowels and consonants they would like. This got me wondering which was the best combination to pick in order to get a nine letter word.

Assuming the letters in countdown are still distributed like this, the probability of getting combinations of letters can be calculated. As the probability throughout the game is dependent on which letters have been picked, I have worked out the probability of getting a nine letter word on the first letters game.

The Probability of YODELLING

YODELLING has three vowels and six consonants. There are 6 (3!) ways in which the vowels could be ordered and 720 (6!) ways in which the consonants can be ordered, although each is repeated at there are two Ls, so there are 360 distinct ways to order the consonants. The probability of each of these is:

21×13×13×6×3×5×4×8×1

67×66×65×74×73×72×71×70×69

So the probability of getting YODELLING is:

6×360×21×13×13×6×3×5×4×8×1	= 0.000000575874154
67×66×65×74×73×72×71×70×69

The Probability of Any Nine Letter Word

I got my computer to find the probability of every nine letter word and found the following probabilities:

Consonants	Vowels	Probability of nine letter word
0	9	0
1	8	0
2	7	0
3	6	0.000546
4	5	0.019724
5	4	0.076895
6	3	0.051417
7	2	0.005662
8	1	0.000033
9	0	0

So the best way to get a nine letter word in the first letters game is to pick five consonants and four vowels.

Pointless Probability

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		1	2	3	4
	1	0	0	1 4	1
	2	0	1 2	3 4	0
	3	3 4	1 2	0	0
	4	1 4	0	0	0

Call the probability of an episode having one, two, three or four new teams P₁, P₂, P₃ and P₄ respectively. After a few episodes, the following must be satisfied:

P1 =	1	P3 + P4
	4
P2 =	1	P2 +	3	P3
	2		4
P3 =	3	P3 +	1	P4
	4		2
P4 =	1	P1
	4

And the total probability must be one:

P1 + P2 + P3 + P4 = 1

These simultaneous equations can be solved to find that:

P1 =	4
	35
P2 =	18
	35
P3 =	12
	35
P4 =	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.