We Got A Problem #12: Tile Shufflin’

Online word games, such as Scrabble, now let you “shuffle” your tiles with the touch of a button.  As far as I can tell, when you click the “shuffle” button the 7 tiles are randomly rearranged into any of the 7! = 5040 different orderings.

And there’s an animation of the tiles moving.  Well… sort of.  Because, sometimes one or more of the tiles doesn’t move, because they are in the same position before and after the shuffling.

How does the probability of having one stuck tile compare to the probability of having no stuck tiles? How does the probability of having one stuck tile compare to the probability of having two stuck tiles? Three? Four?

A Scrabble variant called Lexulous gives players eight tiles instead of seven.  What happens then?

What is the approximate probability of having no stuck tiles?

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The Gambling Machine

The NY Times “Numberplay” blog recently posted an interesting question called “The Gambling Machine”.  The problem was based on a problem found in the NCTM 2011 slides from “Mathematics of Game Shows”, from the PCMI 2007 materials, and originally from the (thankfully cancelled) game show National Bingo Night.

 

The Numberplay article asks about the expected return of proper play in the game, and whether or not a player can get $40 return from the game.  A reasonable guess is for the player to guess higher when the first number is half or less, and guess lower when the first number is more than half, but this is not the best possible strategy.

The article provides some analysis and solutions, and I’ve provided my solution as this PDF.