The probability comes out to be 9/32.
Deducing Arrival Time Possibility
All the possible arrival times by Alice and Bob, in minutes after 5 PM, is constrained by the values of given values of x and y,
x = 0, y = 0, x = 60, y = 60.
Here, let the values of x represent how many minutes after 5 PM Alice arrives at the party.
Let the y values represent the time in minutes that Bob arrives at the party after 5 PM.
Calculating the Required Probability
The times that concern us, however, are obtained by the following probability function,
x + y ≤ 45.
And x = 0, y = 0, and x + y ≤ 45 define the constraints of this probability function. Thus the perimeter of such a graph will be given as,
(45² / 2) = 1012.5 square units
Since the total area of the various arrival times is 60 x 60, or 3600 square units, the probability that Alice and Bob will arrive together after 5 o'clock in the evening in less than 45 minutes is therefore = 1012.5 / 3600 = 9/32
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