Fiendish probability Qs - related to scrabble

yaledo

You've heard the one about the black & white socks in the drawer... you know what are the chances of not picking up the Ace of hearts 5 times in a row... Here's something along vaguely similar lines which is surprisingly complex... even to define the questions is tricky...
If you can get question 3 - that's further than I did... If you get question 6 or 7 or 8, then you need to get out more (and that says a lot coming from me!)

It's the start of a scrabble game between me and Michael. There is no time-limit on the game.

Michael to play first
He passes
I pass
He passes
I pass
He passes, and he tells me that he will continue to pass until I play something. In fact, I think that this cannot be an optimal strategy for a strong player, but it is certainly permitted within the rules, and it could certainly be an optimal strategy for a particular player.

For the purposes of this question, We make three assumptions:
1. Michael will remain true to his word, and he will continue to pass until I place some tiles on the board.
2. We have an unlimited amount of time to play this game.
3. Picking a tile from the bag gives us a perfectly random selection.

Also, the following 2 pieces information will be useful in order to answer some questions:
The distribution of letters in a game of scrabble is as follows:
* 2 blank tiles (scoring 0 points)
* 1 point: E ×12, A ×9, I ×9, O ×8, N ×6, R ×6, T ×6, L ×4, S ×4, U ×4
* 2 points: D ×4, G ×3
* 3 points: B ×2, C ×2, M ×2, P ×2
* 4 points: F ×2, H ×2, V ×2, W ×2, Y ×2
* 5 points: K ×1
* 8 points: J ×1, X ×1
* 10 points: Q ×1, Z ×1

The procedure for an exchange is:
1. Place the tile(s) to be exchanged face down on the table
2. Draw the replacement tile(s) from the bag, and place it/them on your rack.
3. Put the tile(s) to be exchanged back into the bag.


At first glance, without knowing what tiles he has on his rack, my optimum strategy here is to exchange my tiles repeatedly until I get the letters IJKMSUZ - allowing me to play 'MUZJIKS' for 128 points.
If I knew what letters he had on his rack, It might be better for me to try for the slightly lower scoring 'QUARTZY', 'TZADDIQ', or a whole host of other words instead.

So looking closer, the optimal strategy is in fact to exchange tiles repeatedly in order to try first to get an idea of what letters he has on his rack.

In order to annoy Michael (he deserves it), and for reasons which could be strategically prudent in a real life situation, I decide at first that I will only exchange one tile on each move, and I will not keep a record of what tiles I have seen.

I will keep the same 6 tiles on my rack each time, and exchange the 7th one each time.

Question 1: (easy question)
If I do not keep track of what tiles I have seen while exchanging, then how many times must I exchange one tile without getting a Q until I can establish with 99% confidence that Michael has a Q?

Question 2: (I think it's as easy as Q1, but It's a little bit more confusing)
If I do not keep track of what tiles I have seen while exchanging, then how many times must I exchange one tile without getting a blank until I can establish with 99% confidence that Michael has both blanks?

Question 3: (fiendishly hard question)
If I do keep track of what letters I have seen while exchanging, then on average, how many times must I exchange one tile until without getting a Q until I can establish with 99% confidence that Michael has a Q?

Question 4: (similarly fiendish to Question 4)
If I do keep track of what letters I have seen while exchanging, then on average, how many times must I exchange one tile without getting a blank until I can establish with 99% confidence that Michael has both blanks?

Question 5:
On average, if I do not limit myself to exchanging a single letter at a time, and I do keep track of what tiles I have seen, how long would it take me to say with 99% confidence that Michael has the Q.

Question 6: (be careful what assumptions you make)
On average, if I do not limit myself to exchanging a single letter at a time, how long will it take me to say for certain that Michael has no E's on his rack. What is the optimal strategy to obtain this information in as few turns as possible. How long should I keep going for before I become 99% confident that he does in fact have an E on his rack?

Question 7:
In general, If I do not limit myself to exchanging a single letter at a time, and I do make a note of what tiles I have seen, to what extent is it possible for me to say for sure what tiles Michael has on his rack? Are there particular racks he could have which would be possible or impossible to determine for sure? What is the rule to say whether it is possible for us to determine exactly what is on his rack?

Question 8:
Generally, what is the optimal strategy for me to follow in order to get as much information as possible about his rack? (If it is impossible to know a particular detail for sure, then assume that I want to be 99% confident of that particular detail)

WexfordMusings

One problem with your scenario, when both players have passed 3 times in a row, the game is over...

MathsManiac

1.) How an earth do you expect us to comment on Michael's rack if you don't show us a photo?
2.) Given the high probability that Micheal, (by virtue of being called Michael,) is in fact male, there's an extremely high probability that his rack is not at all impressive.