Final answer:
The expected waiting time for Annie or Alvie who arrives first is calculated using the expected value of the absolute difference between their arrival times. On average, the waiting time is 15 minutes.
Step-by-step explanation:
The question relates to expected waiting time between two independent events, where Annie and Alvie are arranging to meet for lunch within a certain time window. The question asks for the expected amount of time that the person who arrives first will have to wait for the other, given that arrival times are independent and uniformly distributed within the hour.
Since both Annie (x) and Alvie (y) can arrive at any time between noon and 1:00 p.m. and their arrival times are independent and uniform, we can calculate the expected waiting time by integrating over the possible arrival times.
Steps to Solve:
Set up the integral for the expected value of the absolute difference |x-y|, assuming x,y are distributed uniformly on [0,1], representing the time in hours after noon.
- Compute the integral which will yield the expected waiting time in hours.
- Convert the obtained value in hours to minutes by multiplying by 60, as there are 60 minutes in an hour.
The expected waiting time is found by calculating E[|x-y|], which through mathematical integration gives 1/4 hour or 15 minutes. So, on average, one person would have to wait for 15 minutes.