Investigation 13: Random Rendezvous (due Thursday, November 4)

You may work with one other person on this assignment, handing in one report with both names.  Word-processed reports are preferred to hand-written ones.  Please copy/paste Minitab output into a Word file as appropriate.

Reconsider the “random rendezvous” example from the day26 handout.  Recall that Tom and Mary arrive independently at a restaurant at a random time that is uniformly distributed between 0 and 60 minutes after noon.  Suppose that the first to arrive agrees to wait fifteen minutes for the other to show up.

a) Use Minitab to simulate 1000 arrival times for each person.  Produce a scatterplot of their arrival times, using different symbols for cases in which they meet and cases in which they do not.  [Please be sure that your axes are labeled well.  You can submit the plot that you created in class.]

b) Use this simulation analysis to approximate the probability that they will meet.

c) Use geometry to determine the exact probability that they will meet in this scenario.  [You can summarize/reproduce your analysis from class.]

Now suppose that they change the waiting rules: Tom agrees to wait for 20 minutes, but Mary only agrees to wait for 10 minutes.

d) Use your simulation results to approximate the probability that they will meet in this scenario. Include a scatterplot using different symbols for cases in which they meet and cases in which they do not. [Hint: First determine what must be true of the random variable T-M in order for them to meet.]

e) Use geometry to determine the exact probability that they will meet in this scenario.  Is the probability larger, smaller, or the same as in the original scenario?

Now suppose that Tom’s arrival time is uniformly distributed between 0 and 45 minutes after noon, and suppose that Mary’s arrival time is uniformly distributed between 15 and 60 minutes after noon. Return to the rule that each waits 15 minutes for the other to show up.

f) Again use Minitab to simulate 1000 arrival times for each person and to produce a scatterplot of their arrival times, using different symbols for cases in which they meet and cases in which they do not.  Comment on how this scatterplot differs from that in a).

g) Use this simulation analysis to approximate the probability that they will meet.  Does the probability appear to be larger, smaller, or the same as in the original scenario?

h) Use geometry to determine the exact probability that they will meet in this scenario.  Is the probability larger, smaller, or the same as in the original scenario?