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## Sunday, 6 April 2014

### 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.