Re: [math-fun] Multiply-gendered humanitarians
Dan H. writes: << ... We assume that each [Bingo] board is equally likely, though that can't be true--with over 552 septillion possible boards, only a small fraction ... have ever been printed! ...
Certainly only a small fraction have ever been printed, but that doesn't exclude the possiblity that they are random. Chances are they aren't, but it wouldn't be that hard to hook up a randomizer to the printer so that each board is the result of the appropriate randomization process. (Hmmm, you may have suggested a likely way to win at Bingo in the long run...) << There are at most five equivalence classes of lines: center horizontal(1), off-center horizontal(4), center vertical(1), off-center vertical(4), and diagonal(2). Is the distribution for diagonals the same as for the center horizontal?
If the only question is Find the probability P(X;n) that line X is filled after n draws, then there are only 2 relevant categories for X: those that need 5 matching draws, and those that need only 4. The number of possible sequences of n draws is D_n = 75! / (75-n)!, and given X, the number of winning sequences among these for one of the 8 lines X needing 5 matches (for n >= 5) is W_5 = 5! * ( n! / (5! (n-5)!) ) = n! / (n-5)!, so the winning probability is the quotient W_5 / D_n = P_5 = n! (75-n)! / (75! (n-5)!) = n(n-1)(n-2)(n-3)(n-4) / (75*74*...*(75-n+1)), and likewise for one of the 4 lines X needing only 4 matches, W_4 / D_n = P_4 = n(n-1)(n-2)(n-3) / 75*74*...*(75-n+1). --Dan A.
Hi Everyone, Drifting a bit in topic header - about Bingo cards... About 11 years ago a printing company hired me to help them develop some Bingo card "perms" (permutations) as they are called. Those are selections of numbers which can then be fed thru the printing process to produce cards. Some people with a mathematical bent make some of their income via the design of Bingo perms. I've never played Bingo but understand that there are special games - normal Bingo is 5 in a row, then there is a need-both-diagonals variant, a large C variant (need top and bottom rows and left column), and large box variant (top and bottom, and left and right). Bingo cards are often printed in triplets, that is three cards which use all 75 numbers. As you know col B has numbers 1-15, col I has 16-30 etc. It is considered "bad" to have all evens or all odds in a row or column, so you have fewer choices for each. It is also considered "good" for distinct cards in a perm to differ by at least 4 numbers from one another. There are not really septillions which are sufficiently different from one another to be acceptable for use in a Bingo parlour. There's more but that may give you the flavour of the production of actual Bingo cards. I didn't stay with the assignment very long. The math ran ahead of the programming. I showed that one of their desires, for a perm which was maximal in a certain sense, could not be achieved - and so they decided that their existing ad hoc perms from random generate-test-discard programs were maybe not so bad after all! Actually, I didn't have to "show" anything except to myself. A simple statement, in a definitive tone of voice, that such and such had been proved to be impossible was sufficient to satisfy the client. A three sentence execution of the golden goose. Bottom line: If you want to be a mathemagician, add some fuzz and mystery to your pronouncements. Ken R.
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Ken Roberts