Final answer:
The probability that neither friend will wait more than ten minutes for the other if they plan to meet between 6:00 and 6:30 pm is 25%. This is calculated by considering the overlap of their arrival times where their wait would be less than ten minutes and multiplying the individual probabilities of each friend arriving during their respective time slots.
Step-by-step explanation:
The question at hand concerns itself with the probability that two friends will not have to wait more than ten minutes for one another if they agreed to meet between 6:00 and 6:30 pm. To solve this, we must look at the possible time intervals and overlay them such that neither friend waits more than ten minutes. If we sketch a number line representing the time from 6:00 to 6:30 pm, we can mark two intervals where each friend can arrive to satisfy the condition: one starting from 6:00 to 6:20 pm, and the other from 6:10 to 6:30 pm. The overlap between 6:10 and 6:20 pm represents the period where both friends would arrive without waiting for more than ten minutes.
The entire meeting window is 30 minutes, so the length of time either friend can arrive is 20 minutes. Since both conditions need to be met—neither waiting more than ten minutes—the ten-minute window of overlap (from 6:10 to 6:20 pm) is the time frame we are interested in. To find the probability, we take the duration of the overlap (which is 10 minutes) and divide it by the total possible time for arrival (which is 20 minutes for each friend), giving us ½.
However, since there are two independent events—each friend arriving—we must square the probability (since the probability of both independent events occurring is the product of their separate probabilities), resulting in (½) × (½) = ¼ or 25% probability that neither friend waits more than ten minutes.