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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.

  • 1
    In general I think all-vowel hands come up because we don't use our vowels as quickly as we use our consonants. I think it's very rare that anyone actually draws a all-vowel hand on the first turn. But if you tend to play 2 consonants to every vowel it won't take long before you reach an all-vowel hand (Typical since it's easy to play off your opponents vowels). Furthermore every time you draw a consonant thereafter, your temptation is to play it instantly so you get stuck in this position. Most of my friends rarely (if ever) use the exchange option. I use it more than anyone else I know.2012-08-04
  • 0
    I'm not too happy about the second-to-last assumption. It changes the question from "What are the chances of three consecutive hands of all vowels?" to "What are the chances... given that the opponent receives at least two consonants and two vowels on their first turn, and at least four each over two turns?" Michael Lugo's answer is quite elegant, but it only works without this assumption.2012-08-04
  • 0
    +1 for taking a real world problem and making it specific enough to have an answer.2012-08-04

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