A rectangular array of mn numbers arranged in n rows, each consisting of m columns, is said to contain a saddlepoint if there is a number that is both the minimum of its row and the maximum of its column. For instance, in the array 1 3 2 0 − 2 6 .5 12 3 the number 1 in the first row, first column is a saddlepoint. The existence of a saddlepoint is of significance in the theory of games. Consider a rectangular array of numbers as described previously and suppose that there are two individuals— A and B—who are playing the following game: 4 is to choose one of the numbers 1, 2,. .., n and B one of the numbers 1, 2,. . ., m. These choices are announced simultaneously, and if A chose i and B chose j. then A wins from B the amount specified by the number in the i th row, j th column of the array. Now suppose that the array contains a saddle point—say the number in the row r and column k call this number x r k . Now if player A chooses row r, then that player can guarantee herself a win of at least x r k (since x r k is the minimum number in the row r). On the other hand, if player B chooses column k, then he can guarantee that he will lose no more than x r k (since x r k is the maximum number in the column k). Hence, as A has a way of playing that guarantees her a win of x r k and as B has a way of playing that guarantees he will lose no more than x r k it seems reasonable to take these two strategies as being optimal and declare that the value of the game to player A is x r k . If the nm numbers in the rectangular array described are independently chosen from an arbitrary continuous distribution, what is the probability that the resulting array will contain a saddlepoint?
A rectangular array of mn numbers arranged in n rows, each consisting of m columns, is said to contain a saddlepoint if there is a number that is both the minimum of its row and the maximum of its column. For instance, in the array 1 3 2 0 − 2 6 .5 12 3 the number 1 in the first row, first column is a saddlepoint. The existence of a saddlepoint is of significance in the theory of games. Consider a rectangular array of numbers as described previously and suppose that there are two individuals— A and B—who are playing the following game: 4 is to choose one of the numbers 1, 2,. .., n and B one of the numbers 1, 2,. . ., m. These choices are announced simultaneously, and if A chose i and B chose j. then A wins from B the amount specified by the number in the i th row, j th column of the array. Now suppose that the array contains a saddle point—say the number in the row r and column k call this number x r k . Now if player A chooses row r, then that player can guarantee herself a win of at least x r k (since x r k is the minimum number in the row r). On the other hand, if player B chooses column k, then he can guarantee that he will lose no more than x r k (since x r k is the maximum number in the column k). Hence, as A has a way of playing that guarantees her a win of x r k and as B has a way of playing that guarantees he will lose no more than x r k it seems reasonable to take these two strategies as being optimal and declare that the value of the game to player A is x r k . If the nm numbers in the rectangular array described are independently chosen from an arbitrary continuous distribution, what is the probability that the resulting array will contain a saddlepoint?
Solution Summary: The author explains how the probability of a saddle point in an array of size n is calculated by using the following expression:
A rectangular array of mn numbers arranged in n rows, each consisting of m columns, is said to contain a saddlepoint if there is a number that is both the minimum of its row and the maximum of its column. For instance, in the array
1
3
2
0
−
2
6
.5
12
3
the number 1 in the first row, first column is a saddlepoint. The existence of a saddlepoint is of significance in the theory of games. Consider a rectangular array of numbers as described previously and suppose that there are two individuals— A and B—who are playing the following game: 4 is to choose one of the numbers 1, 2,. .., n and B one of the numbers 1, 2,. . ., m. These choices are announced simultaneously, and if A chose i and B chose j. then A wins from B the amount specified by the number in the
ith row, jth column of the array. Now suppose that the array contains a saddle point—say the number in the row r and column k call this number
x
r
k
. Now if player A chooses row r, then that player can guarantee herself a win of at least
x
r
k
(since
x
r
k
is the minimum number in the row r). On the other hand, if player B chooses column k, then he can guarantee that he will lose no more than
x
r
k
(since
x
r
k
is the maximum number in the column k). Hence, as A has a way of playing that guarantees her a win of
x
r
k
and as B has a way of playing that guarantees he will lose no more than
x
r
k
it seems reasonable to take these two strategies as being optimal and declare that the value of the game to player A is
x
r
k
. If the nm numbers in the rectangular array described are independently chosen from an arbitrary continuous distribution, what is the probability that the resulting array will contain a saddlepoint?
Patterns in Floor Tiling A square floor is to be tiled with square tiles as shown. There are blue tiles on the main diagonals and red tiles everywhere else.
In all cases, both blue and red tiles must be used. and the two diagonals must have a common blue tile at the center of the floor.
If 81 blue tiles will be used, how many red tiles will be needed?
For what numbers in place of 81 would this problem still be solvable?
Find an expression in k giving the number of red tiles required in general.
At a BBQ, you can choose to eat a burger, hotdog or pizza. you can choose to drink water, juice or pop. If you choose your meal at random, what is the probability that you will choose juice and a hot dog? What is the probability that you will not choose a burger and choose either water or pop?
a card is drawn from a standard deck of 52 cards. If a card is choosen at random, what is the probability that the card is a)heart b)a face card or c)a spade or 10
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