College Algebra, Chapter 10, 10.5, Section 10.5, Problem 16

A safe containing $\$ 1,000,000$ is locked with a combination lock. You pay $\$ 1$ for one guess at the six digit combination. If you open the lock, you get to keep the million dollars. What is your expectation?

The problem states that the six digit combination lock needs to pen in order. So there are $P(6,6)$ ways to open the six-combination lock. In which one is the correct combination and the others are all incorrect. Thus, the expected winning is


$
\begin{equation}
\begin{aligned}
=& 1,000,000 \left( \frac{1}{P(6,6)} \right) - (1) \left( \frac{P(6,6) - 1}{P(6,6)} \right)
\\
\\
=& 1,000,000 \left( \frac{1}{720} \right) - \left( \frac{719}{720} \right)
\\
\\
=& 1387.89

\end{aligned}
\end{equation}
$


This means that we expect to earn $\$ 1387.89$, on average, every time we try to open the lock.