Scrabble Draw Probability

Food for Thought

The traditional way to calculate probability uses Combination. It's a bit computational but it is essentially based on the number of tiles of a certain type in a set, and the number of tiles drawn. Normally the combination is based on what letter is drawn, so with 12 E's, E is the most probable. A or I would be next, with 9 tiles each. Giving the combinatorial nature of he calculation, it's more likely that a player draws an A and an E, than two E's. The J, Q, X, and Z are the least likely to be drawn. Taking this at face value, the most likely 7 tile draw would include AEIONRT.

But in practice,  another way may be more accurate. In reality,  you are not likely to draw the seven most probable (lowest valued) letters. You will usually get one or more of the higher value letters.   Instead of calculating based on the count of each letter,  what if we divide the set into seven nearly even sets of letters based on letter value.

The first set would include all of the E's and two A's. Then A's and I's; I's, O's, and N's; N's, R's, T's; etc. The last set might include the letters F, H, V, W, Y, K, J, Q, X, Z.

Of course,  which of the letters that will be drawn from a set would depend on the number of that letter in the set.  In the set of 14 tiles containing the J, the chance of drawing from that set is one in seven with the chance of drawing that J is one in fourteen.

While the probability of drawing a single tile remains the same,  it can be helpful to realize that the chance of drawing a tile from the high tile set is one in seven, actually more probable than drawing an E. Drawing seven tiles, the chance approaches one in one in each full draw from the tile bag. That realization could change your study habits and playing/exchange philosophy. Using letter value sets, the most probable draw would include 5 1-point tiles, 1 2-3 point tile, 1 4-10 point tile.