We’ve all been confounded by an all-vowel “hand” in Scrabble. What is the likelihood of it happening 3 turns in a row?
Some assumptions are necessary. Let’s use these:
- Two-player game
- English tile distribution
- Vowels are AEIOU (ie, exclude Y)
- The situation is drawing all vowels on the first pull, followed by being in that situation on the subsequent two turns.
- The subject draws first.
- The opponent plays first (this probably contradicts the game rules but is convenient for this experiment).
- The opponent plays a four-letter word on every turn, and that word is two consonants and two vowels.
- The subject is able to play two of their vowels on each turn, requiring a draw of two new tiles.
(If any of the above assumptions is logically impossible, please correct.)
An interesting bit is that blanks can be counted as either a consonant or vowel.
Distribution of English words and letter usage is disregarded for this calculation. Though if you want to throw it in for extra interest, great.
Best Answer
The identity of the letters doesn't matter, so let's just note that there are 42 vowels, 56 consonants, and 2 blanks.
Imagine that the game starts by putting these tiles in random order, 1 through 100.
The subject draws tiles 1 through 7, and the opponent draws tiles 8 through 14.
The opponent then plays four tiles and draws tiles 15 through 18; the subject plays two tiles and draws tiles 19 and 20. This repeats, with the opponent drawing tiles 21 through 24, and the subject drawing tiles 25 and 26.
So under your assumptions, you're asking for the probability that positions 1 through 7, 19, 20, 25, and 26 are all among the vowels. (Blanks don't count as vowels here; if I had six vowels and a blank I don't think I'd say "I have all vowels", although I might be unhappy with my hand.) This is the probability that, if I draw eleven tiles at random, they will all be vowels.
There are ${100 \choose 11}$ ways to choose eleven tiles; of these ${42 \choose 11}$ consist entirely of vowels. So the answer is ${42 \choose 11}/{100 \choose 11}$ which is about $3 \times 10^{-5}$.
Note that your assumptions on how many tiles the opponent is going to play don't matter at all, but your assumptions on how many tiles the subject is able to play matter quite a bit.