Probability of drawing an A from a full Scrabble bag

Scrabble Letter Frequency

Scrabble Letter Frequency

What is the probability of getting an A on your first draw when playing Scrabble? How do I determine this outcome?

A full Scrabble bag contains 100 tiles. To work out the probability of drawing any given letter from a full bag of tiles you simply count how many copies of that letter there are in a full set of tiles and divide that number by 100.

The letter frequency (i.e. the number of copies of each letter in a Scrabble set) is printed on the outer edge of your Scrabble board. I've attached a picture of this to the top of this page. There you will see that there are nine A-tiles. This means the chances of drawing an A are 9/100 or 9% or 0.09 - whichever format you prefer.

If this issue arose over the draw to determine which person gets to play first, don't forget that a blank tile beats an A. I'll leave it to you to work out the odds of drawing a blank!

I hope that helps,


P.S. Please include your name and location in submitted questions. If a submission does not include these things, I generally don't publish it. A first name is fine, so you don't have to worry about being identified.

Comments for Probability of drawing an A from a full Scrabble bag

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Drawing more than one tile
by: Derek

In my example I consider the simple case of drawing one tile from a full bag. If the bag isn't full (i.e. part way through a game), the reasoning is just the same (as per Trevor's correct example).

Things get a bit trickier when you are drawing more than one letter, and you just want one of them to be an A. It's easiest to explain with an example. Suppose there are 40 tiles left in the bag, and 3 of them are A's. You decide to play off 3 letters in the hope of getting an A. What are the chances that you will succeed?

Believe it or not, it is easier to first work out the probability that you'll fail. That means that you fail to get an A on the first tile drawn (37/40), and again on the second (36/39), and again on the third (35/38). The probability of all these things happening is...

37/40 x 36/39 x 35/38

which is about 0.79, or 79%. So that means the chance that you'll succeed is about 0.21, or 21%. So roughly one in five times you perform this experiment, you'll draw at least one A when you replenish the 3 tiles you played.

Assuming I haven't slipped up of course - I'll recheck this tomorrow to make sure ;-)

Hope that helps.

Derek

P.S. If you find this a bit difficult to follow, don't worry. That's because probability is difficult!
Probabilities continued
by: Trevor

So am I right in assuming that the same formulae can be used to calculate the probability of drawing any tile at any time in the game. eg a calculated 1 tile fish.

I will use the A as an example again.

40 Tiles Left in bag
3 As remaining

Probability = 3 / 40 = 0.075 (7.5%)

Obviously i used the above example using a one tile fish, how would the probablity change if i played a 3 letter word thus further increasing my chances of drawing the A.
Pls clarify
by:

Thanks so much for the feedback. Since each person only gets to pick 7 tiles, wouldn't there be less than a 9% chance of picking an A?

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