Kitri's Variation, Act I
Kitri is once again waiting for Basilio at a crowded Barcelona marketplace. Both are planning on running away together from Kitri's father Lorenzo, an inn-keeper who disdains poor Basilio and wants her to marry nobleman Gamache. Unfortunately, Lorenzo and Gamache are also trying to get to her. Ideally, Kitri wants Basilio to arrive first, then Lorenzo, and Gamache last, because breaking away from Lorenzo is easier than from Gamache. Kitri wants to know how likely this ordering is going to happen. She hopes that the math is not too complicated.
Fortunately, the arrival times of Basilio, , of Lorenzo, , and of Gamache, , have the same exponential distribution with mean . Given that these arrival times are mutually independent, Kitri is able to find , the probability that Basilio comes first, Lorenzo second, and Gamache last. What is ?
Give your answer as , rounded to the nearest integer.
Hint: Unlike her calculating father, the elegant Kitri is not keen on heavy computations.
The answer is 17.
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1 solution
Fortunately for Kitri, she doesn't have to do any heavy computations.
There are six possible arrival orderings (Basilio first, Lorenzo second, Gamache third; Basilio second, Lorenzo first, Gamache third; and so on). Now, since all of them are equally likely (because the arrival times are identically and independently distributed), the probability of each scenario occurring, P , is just one in six.
Therefore, the answer we seek is 1 0 0 P = 6 1 0 0 ≈ 1 7 .
Note that the probability that any two people arriving at the same time is zero, so we don't have to worry about those cases.