jim@ism780.UUCP (Jim Balter) (09/21/83)
If I read another word about Monty Hall, division by zero, multi-valued functions, or "phenomenum"s, I am going to SCREEEAM! (Fortunately, I don't have voice input into the system.) Here is my vote for net.math.high-school. Try these Probability Puzzlers for fun: A card is lost from a deck. Five cards are then dealt from the remaining 51 cards and it is found that all 5 cards are clubs. What is the probability that the lost card is a club? On a panel of three judges, two are "normal" and have a probability of p of judging evidence correctly; the third flips a (balanced) coin to decide. What is the probability that the committee will arrive at a correct decision by taking a majority vote? A box contains 6 cake donuts and 6 glazed donuts. A second box contains 10 cake donuts and 5 glazed donuts. A pair of dice is tossed. If the sum showing on the two dice is exactly divisible by 3, a donut is taken at random from the first box. Otherwise a donut is taken from the second box. What is the probability of obtaining a cake donut? A ball has been drawn at random from a bag containing 99 black balls and 1 white ball. A woman whose statements are accurate 9 times out of 10 asserts that the white ball was drawn. What are the odds that it really was? If a coin were shaken out of a certain piggy bank, its expected value would be 15 cents. If a dime had been added, the expected value would have been only 14 cents. What are the contents of the bank? How many people would you expect to meet, on the average, before you met one who was born on a Sunday? A coin is so unbalanced that you are as likely to get two heads in a row as you are to get a tail in one toss. What is the probability of getting a head in one toss? The squares of a chessboard are numbered randomly, all differently. What is the probability that at least one of the squares is both the maximum in its row and the minimum in its column? If two numbers, not necessarily distinct, are picked at random from the integers 1, 2, ..., n, what is the probability that the sum of the numbers is at most n? The sum of two positive integers is 99. What is the probability that their product is greater than 1000? Smith and Jones, both 50% marksmen, decide to fight a duel in which they exchange alternate shots until one is hit. What are the odds in favor of the one who shoots first? Jim Balter (decvax!yale-co!ima!jim), Interactive Systems Corp --------
egs@ulysses.UUCP (09/26/83)
It would seem that most of Jim's puzzles can be solved by brute force.
The exception is the chessboard puzzle which depends on noticing that it is not
possible to have more than one square whose number is max in its row and min
in its column. For example, if both w and x in this picture:
. .
. .
...w...y...
. .
. .
...z...x...
. .
. .
satisfy the max-min requirement, then we must have w>y, w<z, x>z, and x<y. In
other words, w>y>x and w<z<x, which are contradictory. Having noticed this,
we need merely count the ways in which each number can satisfy the max-min
requirement without having to worry about double counting. First, replace each
number with its rank order (i.e. 1, 2, 3, ..., N on an nxn board with N=n^2).
For a number m from this set, we must multiply the following numbers:
C(m-1,n-1)*(n-1)! number of ways in which we can select n-1
numbers from the set numbers 1,...,m-1
and arrange them on the same column as m
C(N-m,n-1)*(n-1)! number of ways the numbers larger than m can
be put on the same row as m
(N-2n+1)! number of ways the remaining numbers can be
arranged on the remaining squares
N number of squares on which the number m can rest
which gives:
(n!)^2 * ((n-1)^2)! * C(m-1,n-1) * C(N-m,n-1).
Summing these terms over legitimate values of m (from n to N-n+1) and dividing
by N! will give the answer. For n=2, I get 2/3. For n=3, the answer is .3. For a
regular chessboard (n=8), I get .001243 or there abouts.
Ed Sheppard
BTL @ MH