Probability of Two People Meeting Given Random Waiting Times
In this article, we will explore the mathematical probability of two individuals meeting when their time slots are fixed, but they allow a certain waiting duration. We will apply a mathematical model to determine the likelihood of a successful meeting under the conditions given.Conditions and Model Setup
The meeting is arranged between 2:00 PM and 3:00 PM, providing a 60-minute window. The first person, upon arrival, waits for 15 minutes before leaving. Let's denote the arrival time of the first person as ( X ) and the second person as ( Y ), both uniformly distributed between 0 and 60 minutes after 2:00 PM.Conditions for Meeting
For a successful meeting to occur, the arrival time ( Y ) of the second person must satisfy the condition: [ X leq Y leq X 15 ] This constraint can be visualized on a 2D coordinate plane, where the x-axis represents the arrival time of the first person and the y-axis represents the arrival time of the second person.Geometric Representation
Consider the x-axis and y-axis ranging from 0 to 60 minutes. We can now determine the area where the two individuals meet within the given constraints.Triangular Areas:
For ( 0 leq X leq 45 ): - The area of the region where the second person's arrival time falls within the first person's waiting period is a triangle with a base and height of 15 minutes. The area is calculated as: [ text{Area}_{0 leq X leq 45} frac{1}{2} times 15 times 15 112.5 text{ square minutes} ] For ( 45 leq X leq 60 ): - The upper limit for ( Y ) is capped at 60. The area forms a trapezoid with the height of ( 60 - X ) and the bases of 15 and 0 minutes. The area of this trapezoid is: [ text{Area}_{45 leq X leq 60} frac{1}{2} times 15 0 times (60 - 45) frac{1}{2} times 15 times 15 112.5 text{ square minutes} ] Adding both areas provides the total meeting area: [ text{Total Meeting Area} 112.5 112.5 225 text{ square minutes} ]