Question
Two friends decide to meet somewhere between 1pm and 2pm. Both will choose a time randomly, between 1pm and 2pm, and wait for 15 minutes before leaving. What are the odds that they will actually meet?
Check your answer
Solution
Answer: 7/16
Let \( t_a \) denote the time that friend A arrives, and let \( t_b \) denote the time that friend B arrives. Measure both times in minutes after 1pm, so both lie in \( [0,60] \). The friends meet exactly when their arrival times differ by at most 15 minutes:
\[ |t_a - t_b| \le 15 \] The total outcome space is a \( 60 \times 60 \) square. The non-meeting region consists of two right triangles where \( |t_a - t_b| > 15 \), each with side length 45: \[ A = 2 * 45 * 45 * \frac{1}{2} = 2025 \] Thus, the meeting probability is: \[ P = \frac{3600 - 2025}{3600} = \frac{1575}{3600} = \frac{7}{16} \] Therefore, the probability that the two friends will actually meet is \( \frac{7}{16} \).
\[ |t_a - t_b| \le 15 \] The total outcome space is a \( 60 \times 60 \) square. The non-meeting region consists of two right triangles where \( |t_a - t_b| > 15 \), each with side length 45: \[ A = 2 * 45 * 45 * \frac{1}{2} = 2025 \] Thus, the meeting probability is: \[ P = \frac{3600 - 2025}{3600} = \frac{1575}{3600} = \frac{7}{16} \] Therefore, the probability that the two friends will actually meet is \( \frac{7}{16} \).