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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.

Mellie  Apr 30, 2015

Best Answer 

 #1
avatar+87301 
+10

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

 

 

  

CPhill  Apr 30, 2015
 #1
avatar+87301 
+10
Best Answer

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

 

 

  

CPhill  Apr 30, 2015

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