Final answer:
The probability that neither friend needs to wait more than ten minutes for the other when they meet between 6:00 and 6:30 pm is 4/9, calculated by considering the overlapped area on a coordinate plane representing their possible arrival times.
Step-by-step explanation:
The question deals with finding the probability that neither friend will have to wait more than ten minutes for the other when they meet between 6:00 and 6:30 pm. To solve this problem, we need to use the concept of uniform distribution since the arrival time of each friend is equally likely to be any time within the 30-minute interval.
Let us consider the 30-minute window from 6:00 to 6:30 pm and represent this on a number line. Each point on this number line represents a potential arrival time for the friends. If one friend arrives at a certain time, the other friend must arrive within a 10-minute window following that time to ensure that neither has to wait more than ten minutes. Visually, this scenario creates an overlapping area on the number line where their arrival times will fulfill the condition.
The overlapping area forms a square on a coordinate plane, where each axis represents the arrival times of the two friends between 6:00 to 6:30 pm. The total area of the square represents the total possible combinations of arrival times (30 minutes × 30 minutes = 900 possible minutes). The area where neither friend would have to wait more than ten minutes is represented by a smaller square, shifted by a 10-minute margin from each axis, resulting in an overlap of 20 minutes × 20 minutes = 400 possible minutes.
Thus, the probability that neither friend has to wait more than ten minutes is the ratio of the overlapped area to the total area. This gives us:
Probability = Overlapped Area / Total Area = 400 / 900 = 4 / 9.
Therefore, the probability that neither friend has to wait more than ten minutes for the other is 4/9.