Kitri's Variation, Act II

Probability Level 3

Kitri and Basilio have managed to flee from the hands of Lorenzo and Gamache and arrived at Port de Barcelona. They are now waiting for a ship to pick them up. As a frequent seafarer and an amateur statistician, Basilio knows that the late time $T$ of ships arriving at Port de Barcelona follows an exponential distribution with an unknown parameter $\lambda$ . From his observations, ships never arrive early, but the probability that a ship arrives within the first 15 minutes after scheduled time is $P(T \leq 15) = 0.9$ - quite likely for a busy port.

Unfortunately, Kitri and Basilio have been waiting for their ship for an hour. The impulsive Basilio is growing impatient - he is not hopeful that their ship will arrive within the next 15 minutes. The clever Kitri, on the other hand, remains calm. Given that they have been waiting for $1 \text{ hour}$ , Kitri quickly finds that are they going to wait an $\text{extra } 15 \text{ minutes or more}$ with probability $P$ .

What is $100P$ , rounded to the nearest integer?

The answer is 10.

1 solution

Huan Bui
Dec 31, 2018

This is a straightforward application of the memoryless property of the exponential distribution, with $s = 15$ and $t = 60$ .

$P = P(T > t+s \vert T > t) = P(T>s) = 1-P(T\leq s) = 1-0.9 = 0.1$ . The answer we seek is $100P = 100\times 0.1 = \boxed{10}$ .

Note that $P$ is independent of $t$ , how long they have waited for.

Kitri's Variation, Act II

The answer is 10.

1 solution

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