To determine the probability of you and your friend meeting each other at the restaurant, we need to consider the range of arrival times and calculate the overlapping time intervals.
Given:
- You and your friend arrive at the restaurant at a random time between noon and 1 p.m.
- You both stay for 15 minutes.
To simplify the calculation, let's consider time in minutes, with 0 representing noon and 60 representing 1 p.m.
The possible arrival time for each of you is randomly distributed within this 60-minute range.
To calculate the probability of meeting, we need to determine the overlapping time intervals when both of you are present at the restaurant. There are two scenarios:
1. Your friend arrives during your 15-minute stay:
In this case, your friend can arrive at any time between (your arrival time - 15 minutes) and (your arrival time + 15 minutes). This results in a 30-minute window during which your friend can arrive and meet you.
2. You arrive during your friend's 15-minute stay:
Similarly, you can arrive at any time between (your friend's arrival time - 15 minutes) and (your friend's arrival time + 15 minutes), resulting in another 30-minute window of overlap.
To calculate the total probability of meeting, we need to sum the probabilities of both scenarios:
Probability = Probability of scenario 1 + Probability of scenario 2
Since both scenarios have the same probability, the final probability can be calculated as:
Probability = 2 * Probability of scenario 1
The probability of scenario 1 can be calculated by dividing the 30-minute window by the total time range of 60 minutes:
Probability of scenario 1 = 30 minutes / 60 minutes = 0.5
Therefore, the total probability of meeting each other at the restaurant is:
Probability = 2 * Probability of scenario 1 = 2 * 0.5 = 1
Hence, the probability of you and your friend meeting each other at the restaurant is 1, or 100%.