1(a) Suppose two friends Anjali and Prabhat trying to meet for a date to have lunch say between 2pm2 \mathrm{pm} to 3pm3 \mathrm{pm}. Suppose they follow the following rules for this meeting:
Each of them will reach either on time or 10 minutes late or 20 minutes late or 30\mathbf{3 0} minutes late or 40 minutes late or 50 minutes late or 1 hour late. All these arrival times are equally likely for both of them.
Whoever of them reaches first will wait for the other to meet only for 10 minutes. If within 10 minutes the other does not reach, he/she leaves the place and they will not meet.
Find the probability of their meeting.
Answer:
Introduction
Anjali and Prabhat are trying to meet for lunch between 2 pm and 3 pm. They can arrive at seven different times: on time, 10, 20, 30, 40, 50 minutes late, or 1 hour late. Each of these arrival times is equally likely. The first to arrive will wait for 10 minutes for the other. We need to find the probability that they will meet.
Work/Calculations
Step 1: Define the Sample Space
The sample space SS consists of all possible arrival times for Anjali and Prabhat. Since each of them has 7 choices, the total number of outcomes is 7xx77 \times 7.
|S|=7xx7=49|S| = 7 \times 7 = 49
Step 2: Identify Favorable Outcomes
A favorable outcome is one where they meet. This can happen if:
Both arrive at the same time: There are 7 such cases (0, 10, 20, 30, 40, 50, 60 minutes late).
One arrives and the other arrives within the next 10 minutes: For example, if Anjali arrives 10 minutes late, Prabhat can arrive either on time or 20 minutes late for them to meet.
To count these, we consider the time difference between their arrivals. The time difference can be 0, 10, 20, 30, 40, 50, or 60 minutes. They will meet if the time difference is 0 or 10 minutes.
For a time difference of 0 minutes, there are 7 cases.
For a time difference of 10 minutes, there are 6 cases for Anjali arriving first and 6 for Prabhat arriving first, making it 12 cases.
So, the total number of favorable outcomes |E||E| is 7+12=197 + 12 = 19.
Step 3: Calculate the Probability
The probability PP of them meeting is given by:
P=(|E|)/(|S|)P = \frac{|E|}{|S|}
After calculating, we get the value of PP.
After calculating, we find that the probability of Anjali and Prabhat meeting is (19)/(49)\frac{19}{49}.
Conclusion
The probability that Anjali and Prabhat will meet for lunch, given the conditions, is (19)/(49)\frac{19}{49}. This takes into account all the possible times they could arrive and the 10-minute waiting time.