Final answer:
To find the probability that the person's total waiting time will be between 3.7 and 3.9 minutes, we can subtract the probability that the person waits less than 3.7 minutes from the probability that the person waits less than 3.9 minutes. We find that the probability is 0.2.
Step-by-step explanation:
To find the probability that the person's total waiting time will be between 3.7 and 3.9 minutes, we need to consider the probability that the person waits less than 3.9 minutes. First, we subtract the time the person has already been waiting (2.3 minutes) from 3.9, giving us 1.6 minutes. Since the waiting times follow a uniform distribution from 0 to 8 minutes, the probability that the person waits less than 1.6 minutes is the proportion of the interval [0, 8] that is less than 1.6.
The width of the interval [0, 8] is 8 - 0 = 8 minutes. The width of the subinterval [0, 1.6] is 1.6 - 0 = 1.6 minutes.
The probability that the person waits less than 1.6 minutes is (width of [0, 1.6]) / (width of [0, 8]) = 1.6 / 8 = 0.2.
Therefore, the probability that the person's total waiting time will be between 3.7 and 3.9 minutes is equal to the probability that the person waits less than 3.9 minutes minus the probability that the person waits less than 3.7 minutes.
Let's calculate:
Probability (person waits less than 3.7 minutes) = 1.6 / 8 = 0.2
Probability (person waits less than 3.9 minutes) = 3.2 / 8 = 0.4
Probability (the person's total waiting time is between 3.7 and 3.9 minutes) = probability (person waits less than 3.9 minutes) - (person waits less than 3.7 minutes)
= 0.4 - 0.2 = 0.2