Question

Persons A and B decide to arrive and meet sometime between 7 and 8 pm. Whoever arrives first will wait for ten minutes for the other person. If the other person doesn't turn up inside ten minutes, then the person waiting will leave. What is the probability that they will meet?

• A. $\frac{1}{2}$
• B. $\frac{1}{6}$
• C. $\frac{5}{6}$
• D. $\frac{11}{36}$
• E. $\frac{25}{36}$

Hint:

When dealing with probability problems involving random events, often visualizing the problem as a geometric region (like a square) helps simplify the calculation by determining the area of favorable outcomes.

Answer 1

The Correct Option is D

Step 1: Understanding the scenario.
We are given that two people, A and B, arrive between 7 and 8 pm, with the time interval for their arrival being one hour. They each arrive at a random time between 7 and 8 pm. We can model this scenario using a coordinate plane where: - The x-axis represents the time of arrival of person A (between 0 and 60 minutes). - The y-axis represents the time of arrival of person B (also between 0 and 60 minutes). Person A and B will meet if they arrive within 10 minutes of each other. This means that the time difference between their arrival times must be less than or equal to 10 minutes. The meeting region is represented by the area inside the coordinate system where the difference between the times is at most 10 minutes. Step 2: Calculating the probability.
The total possible outcomes for the two people arriving are represented by the area of the square, which has side length 60 minutes (from 7:00 pm to 8:00 pm), so the total area is: \[ \text{Total Area} = 60 \times 60 = 3600 \] The area where they will meet (where the difference in arrival times is less than or equal to 10 minutes) is represented by the region within the square where: \[ |x - y| \leq 10 \] This region is a band of width 20 centered along the diagonal of the square. The area of this region can be calculated by subtracting the areas of the triangles where the difference in arrival times is greater than 10 minutes from the total area. The area of each triangle (where the difference is greater than 10 minutes) is: \[ \text{Area of one triangle} = \frac{1}{2} \times (60 - 10) \times (60 - 10) = \frac{1}{2} \times 50 \times 50 = 1250 \] Thus, the total area of the two triangles is \( 2 \times 1250 = 2500 \). So, the area where they meet is: \[ \text{Meeting Area} = 3600 - 2500 = 1100 \] Step 3: Probability of meeting.
The probability of them meeting is the ratio of the meeting area to the total area: \[ P(\text{meeting}) = \frac{1100}{3600} = \frac{11}{36} \] Step 4: Conclusion.
Thus, the probability that A and B will meet is \(\frac{11}{36}\), which corresponds to (D).