How often do random tiles spell a word?
Reach into a bag of Scrabble tiles, pull out seven at random, and lay them down in a row in the order they came out. What are the odds they spell a real word? Take a guess before you read on. The answer comes from multiplying a chain of fractions, then adding up a lot of tiny numbers.
What's in the bag
A standard English set has 100 tiles: 98 letters and 2 blanks. A blank can stand for any letter. The letters aren't spread evenly. There are 12 E's, 9 A's and 9 I's, down to just one each of J, K, Q, X and Z. That uneven mix is what drives the whole calculation.
A row only counts if it spells a word exactly as the tiles landed, in that order. The words come from ENABLE, a standard list of English words.
The odds of one word
Start with one specific seven-letter word, like ALGEBRA. The first tile has to be an A. There are 9 A's among the 100 tiles, so that's 9 in 100. The second tile has to be an L. There are 4 L's among the 99 tiles left, so 4 in 99. Then a G is 3 in 98, an E is 12 in 97, and so on. By the last A, one A is already gone, so it's 8 A's out of the 94 tiles left.
Each step has to happen, so you multiply them together.
9/100 × 4/99 × 3/98 × 12/97 × 2/96 × 6/95 × 8/94 ≈ 1 in 650 million
What the blanks add
The blanks give every word extra ways to happen. At any spot in ALGEBRA, the tile could be the right letter or one of the blanks still in the bag. Counting every one of those paths improves the odds for ALGEBRA to about 1 in 105 million, around six times better.
Across all words, the effect is big. With seven tiles, keeping the 2 blanks in the bag makes spelling some word about 4 times as likely as taking them out.
Adding up every word
Now repeat that for every seven-letter word in the list. There are 23,109 of them. Add up all their chances, counting each possible row only once even if a blank lets it fit more than one word, and you get the chance that your row spells anything at all. Adding works here because a row can only land one way, so the chances of different rows never overlap.
7 tiles: about 1 in 6,344
That's a surprisingly long shot for something with 23,109 targets. Each word is so unlikely on its own that even thousands of them don't add up to much. The likeliest seven-letter word, worked out over the full bag, is TAENIAE, at about 1 in 26 million. It's built from common tiles like E, A, I, N and T, which is why the words most likely to land are ones almost nobody knows.
Fewer tiles, more tiles
Change the number of tiles and the odds swing hard.
- 2 tiles spell a word about 1 in 3 times.
- 3 tiles: about 1 in 6.
- 5 tiles: about 1 in 114.
- 7 tiles: about 1 in 6,344.
- 10 tiles: about 1 in 19 million.
Short words are easy because there are lots of them and each needs only a couple of lucky draws. Every extra tile adds another fraction to the chain, and each fraction is small, so the odds fall faster than the number of words can make up for. Going from 7 tiles to 10 makes a word about 3,000 times harder to spell.
You can test all of this in the game at the top of the page. Pick how many tiles to pull, turn the blanks on or off, and dump them as often as you like. Or dump 1,000 times at once and compare how many words you get with what the math expects.