Advanced Operations, Statistical Modeling, and Probabilistic Inference

IB AA SL: core and optional content

Core: Cindy arrives before June. Optional: random departure times, total stays and whether the visits overlap.

Two friends June and Cindy always meet in the library to study together. Their arrival time to the library and length of stay in the library is independent of each other. June’s arrival time follows a normal distribution with mean arrival time at \(1700\) h with a standard deviation of \(30\) minutes and her stay in the library follows a normal distribution with mean \(40\) minutes and standard deviation \(15\) minutes. On the other hand, Cindy will reach the library every day at \(1730\) h sharp and her stay in the library follows a normal distribution with mean \(1\) hour and standard deviation \(15\) minutes.

1. On a randomly chosen day, calculate the following probabilities:
   1. Cindy is in the library earlier than June. [2]
   2. June leaves before Cindy arrives. [2]
2. Calculate the probability that the total time spent by June for three days in the library is more than three times the time spent by Cindy in a day. [2]

For a particular week, Cindy is not consistent in turning up at the library. The probability that Cindy will visit the library on any particular day is \(0.7\). If she visits the library, she will turn up at \(1730\) h sharp and her stay will follow the same distribution as before.

3. Show that the probability that June and Cindy will not meet in a particular day is \(0.5705\). [1]