+1836 Alice and Bob both go to a party which starts at 5:00. Each of them arrives at a random time between 5:00 and 6:00. What is the probability that the number of minutes Alice is late for the party plus the number of minutes Bob is late for the party is less than 45? Express your answer as a common fraction.
+129852 Look at the following graph, Mellie........https://www.desmos.com/calculator/9p3hhbrwl2
The graph of all the possible arrival times - in minutes after 5 PM - by both Alice and Bob is bounded by x = 0, y = 0 x= 60 and y = 60.
Let the x values in the graph be the number of minutes after 5 PM that Alice arrives at the party. And let the y values be the number of minutes after 5 PM that Bob arrives at the party. For example, at (0,0), both arrive at 5PM and at (60,60), both arrive at 6 PM. At (0, 45)....Alice arrives at 5PM and Bob arrives at 5:45 PM. At (22.5, 22.5), both arrive at 5:22:30. And at (45, 0), Alice arrives 45 minutes after 5 PM and Bob arrives exactly at 5 PM.
But, the times we are interested in lie beneath the graph of x + y ≤ 45.
And this area is bounded by x = 0, y = 0 and x + y ≤ 45 . And it equals 45^2 / 2 = 1012.5 sq units
Note that the area of the total possible arrival times = 60 x 60 = 3600 sq units
So....the probabilty that the sum of Alice's and Bob's arrival times after 5PM are less than 45 minutes =
1012.5 / 3600 = 9/32
+129852 Look at the following graph, Mellie........https://www.desmos.com/calculator/9p3hhbrwl2
The graph of all the possible arrival times - in minutes after 5 PM - by both Alice and Bob is bounded by x = 0, y = 0 x= 60 and y = 60.
Let the x values in the graph be the number of minutes after 5 PM that Alice arrives at the party. And let the y values be the number of minutes after 5 PM that Bob arrives at the party. For example, at (0,0), both arrive at 5PM and at (60,60), both arrive at 6 PM. At (0, 45)....Alice arrives at 5PM and Bob arrives at 5:45 PM. At (22.5, 22.5), both arrive at 5:22:30. And at (45, 0), Alice arrives 45 minutes after 5 PM and Bob arrives exactly at 5 PM.
But, the times we are interested in lie beneath the graph of x + y ≤ 45.
And this area is bounded by x = 0, y = 0 and x + y ≤ 45 . And it equals 45^2 / 2 = 1012.5 sq units
Note that the area of the total possible arrival times = 60 x 60 = 3600 sq units
So....the probabilty that the sum of Alice's and Bob's arrival times after 5PM are less than 45 minutes =
1012.5 / 3600 = 9/32
