Hi! I'm trying to make a card game (spinoff of the Junkyard Wars game) and I'm trying to determine the number of points I should give each letter.

The premise is this. The number of points for each letter depends on the tile distribution. If there are N of each letter in the deck (well, I'm saying N+1 since there are two blanks and I'm figuring each player will get one), the expected number of tiles we need to draw until we get one of them is 1 / (N+2) since there are N+2 gaps, presumably of equal size on the average, between the ends of the tile distribution and the letters drawn. So far so good.

Now here's the tricky part. Suppose there's are T (T's) and H (H's). The goal is to to get a T and an H. The upper limit is the sum of the expected values for the T and the expected value for the H (which is what I'm using right now -- I'm assuming it's slightly more difficult than the actual calculation because a blank may only be used once). So far so good.

Welll -- I'm now thinking of adding dipthongs. Say, "TH".

There are now TWO ways to get "TH" in your hand: T and H, or TH. The TH would do no good if you just want a T in your hand (which is probably the majority of the cases, so the formula to score the T won't change). But what should the score of the "TH" be?

Thanks in advance,

ACG

There are a bunch of words with T (CAT, for instance), some with H (HORSE)'s, and a few with TH's (THAN). The goal is to spell out the words with the stuff on the cards (which may include dipthongs).