What is the probability of a Scrabble player making two seven letter words on their first two turns?
by Jeremy Lewis
(Nova Scotia, Canada)
Hi Jeremy - thanks for a very interesting question.
First let me explain why there is no 'exact' answer to your question, and then I'll give you a realistic ball-park figure.
The probability of a Scrabble player making a bingo (that's what many players call it when you use up all seven of your letters) depends on...
1. The Scrabble dictionary being used
2. The players' word knowledge, anagramming skills, and strategic styles
So let me make a couple of assumptions...
Assumption 1: The Scrabble dictionary being used is the Official Scrabble Players Dictionary used in North America (which seems reasonable since you are from Canada, after all).
Assumption 2: Both players have perfect word knowledge, perfect anagramming skills, and follow a conventional Scrabble strategy. (Without this assumption, you just can't go anywhere really.)
In that case, the probability that the first player can play a bingo from their first rack can be computed exactly using the known distribution of letters in a standard tile bag, along with the known list of allowable words. It turns out to be about 15%. (That surprises many lounge-room players!)
Now, the probability of getting a bingo on the second move is not possible to work out exactly, since it depends on what the other player decides to play in response to the opening bingo.
However, since on the second move a player has additional bingo opportunities (i.e. they can now play eight-letter words through the opponent's letters), the likelihood of another bingo is probably a little higher than the first - let's say 20% (which I suspect overestimates things a little).
Multiplying these values together gives a probability of about 3%. In other words, if two 'perfect' players were to play each other in 100 Scrabble games, you would expect that in about three of those games the opening player will play two consecutive bingos.
The only way to get a more accurate answer is to have a Scrabble program play itself thousands and thousands of times, which several serious gamers do from time to time. If I can get access to the results of such a simulation, I'll update this answer and we can see how far off my estimate is.
I suspect the result would be a little less than 3%, since my 20% figure is probably an over-estimate.
In the meantime, thanks again for the interesting question Jeremy. And feel free to comment below if you'd like me to follow up on anything.
P.S. Even though your question refers to seven-letter words, I've assumed you really mean bingos, which can be eight letters long, or even longer.
P.P.S. Was this question triggered by someone doing this to you? ;-)