Kitri and Basilio's New Year's Eve date

Probability Level 4

Four days ago, Kitri and Basilio agreed to meet up at a Barcelona marketplace to watch the New Year fireworks. Now, Kitri has arrived on time and been waiting for her beloved Basilio, who is almost never on time. She wonders what the chances are that Basilio arrives more than $45 \text{ minutes}$ late. As much as Kitri loves Basilio, she knows little about his habits, except for the fact that Basilio is always late on average $30 \text{ minutes}$ , give or take $5 \text{ minutes}$ .

Knowing nothing else but a well-known inequality, Kitri works out that a very good upper bound for the probability Basilio arriving later than $45 \text{ minutes}$ is $P$ , and she's somewhat relieved. What is $100P$ , rounded to the nearest integer?

Note: This problem has been modified from a previous version to be more precise.

The answer is 10.

1 solution

Huan Bui
Dec 30, 2018

This is a straightforward application of the one-sided Chebyshev's inequality, with $\mu = 30, \sigma = 5, c = 45-30=15$ :

$P(T \geq \mu + c) = P(T \geq 30+15) \leq \frac{\sigma^2}{\sigma^2 + c^2} = \frac{5^2}{5^2 + 15^2} = \frac{1}{10}$ . So the answer we seek is $100P = \boxed{10}$ .

Thanks @Otto Bretscher for the discussion!

Kitri and Basilio's New Year's Eve date

The answer is 10.

1 solution

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