Formula Test

Example 7.3.5

A person has four keys and only one key fits to the lock of a door. What is the probability that the locked door can be unlocked in at most three tries?
Solution

Let U be the event that the door has been unlocked and L be the event that the door has not been unlocked. We illustrate with a tree diagram.

\documentclass{article}
\usepackage{tikz}

\begin{document}

\end{document}

First Try:

Second Try:

Third Try:

The probability of unlocking the door in the first try = 1/4.
The probability of unlocking the door in the second try = (3/4)(1/3) = 1/4.
The probability of unlocking the door in the third try = (3/4)(2/3)(1/2) = 1/4.
Therefore, the probability of unlocking the door in at most three tries = 1/4 + 1/4 + 1/4 = 3/4.