Linear Programming/Systems of Inequalities
The photocopying machine in a school office is made available to teachers between the hours of 3pm and 4pm. Mr. Grim and Mrs. Grump each have 10 minutes of copying to do each day. If theyh each enter the office at points in the available hour, what is the probability that one of them will have to wait while the other finishes copying?
Hi Sam. Welcome to Jiskha!
Draw yourself an x,y plot with both axes running from 0 to 50 minutes, representing the times that Grim (x) and Grump (y) might enter the room, measured from 3 PM. Assume those times are randomly distributed. The area of (x,y) space where |x-y| < 10 is the region where one or the other person is going to have to wait.
The probability that one of them will have to wait is the ratio or the area of a diagonal region between the lines x = y + 10 and x = y - 10, to the area of the 50 x 50 square. I get that ratio to be
1 - (2*0.5*40*40)/(50*50) = 1 - 16/25 = 9/25
The number (2*0.5*40*40)/(50*50) = 16/25 is the fraction of x,y space where no waiting is required, and is the ratio of the sum or two right-triangular areas to that of the 50 x 50 square
To calculate the probability that one of them will have to wait while the other finishes copying, you can follow these steps:
1. Draw an x-y plot with both axes running from 0 to 50 minutes, representing the times that Mr. Grim (x) and Mrs. Grump (y) might enter the room, measured from 3 PM.
2. The area of (x,y) space where |x-y| < 10 represents the region where one or the other person will have to wait. This is because if the time difference between their entry is less than 10 minutes, one person will have to wait for the other to finish copying.
3. Calculate the ratio of the area of this diagonal region between the lines x = y + 10 and x = y - 10 to the area of the 50 x 50 square.
4. The probability that one of them will have to wait is equal to 1 minus this ratio.
Using the formula, the ratio can be calculated as:
(2 * 0.5 * 40 * 40) / (50 * 50) = 16/25
Therefore, the probability that one of them will have to wait is:
1 - 16/25 = 9/25
So, the probability that one of them will have to wait while the other finishes copying is 9/25.