Question

Two sisters, Allison and Teri, have agreed to meet between 1 and 6 P.M. on a particular day. In fact, Allison is equally likely to arrive at exactly 1 P.M., 2 P.M., 3 P.M., 4 P.M., 5 P.M., or 6 P.M. Teri is also equally likely to arrive at each of these six times, and Allison's and Teri's arrival times are independent of one another. Thus there are 36 equally likely (Allison, Teri) arrival-time pairs, for example, \((2,3)\) or \((6,1)\). Suppose that the first person to arrive waits until the second person arrives; let \(w\) be the amount of time the first person has to wait. a. What is the probability distribution of \(w\) ? b. How much time do you expect to elapse between the two arrivals?

Short answer

The probability distribution of \(w\) is \(6/36, 10/36, 8/36, 6/36, 4/36, 2/36\) for waiting times 0,1,2,3,4,5 hours respectively. The expected waiting time or elapsed time between the two arrivals is given by \(E(w) = 0*(6/36) + 1*(10/36) + 2*(8/36) + 3*(6/36) + 4*(4/36) + 5*(2/36)\)

Step-by-step solution

Identifying the Time Pairs and Possibilities

There are a total of 6 possible arrival time for each sister making a total of \(6*6=36\) arrival time pairs.

Calculate the Possible Waiting Times

The possible waiting times (in hours) can be 0, 1, 2, 3, 4, and 5. The waiting time is 0 if and only if both arrive at the same time. So, the frequency for 0 waiting time is 6 (considering times such as 1 and 1, 2 and 2, etc.) Since, for a waiting time of 1, 2, 3, 4, 5 hours, there are two possibilities: either Allison or Teri comes first. Therefore, the waiting time frequencies for 1,2,3,4,5 can be calculated as \(2(5,4,3,2,1)= 2(15) = 30\) respectively.

Construct the Probability Distribution table

The probability distribution of \(w\) will be calculated considering the frequency of waiting time over total number of outcomes. Given the waiting times as 0,1,2,3,4,5 hours and their respective frequencies 6,10,8,6,4,2, the corresponding probabilities would be \(6/36, 10/36, 8/36, 6/36, 4/36, 2/36\) respectively.

Determine Expected Time of Arrival

The expected time of arrival is calculated as the sum of product of each waiting time and its corresponding probability. Set \(E(w)\) as the expected time of arrival, hence \(E(w) = 0*(6/36) + 1*(10/36) + 2*(8/36) + 3*(6/36) + 4*(4/36) + 5*(2/36)\)