176. The textbook is published in a run of 100,000 copies. The probability that a textbook is improperly bound is 0.0001. Find the probability that the run contains exactly five defective books.
Solution. According to the condition, $n=100000, p=0.0001, k=5$. The events consisting of books being incorrectly bound are independent, the number $n$ is large, and the probability $p$ is small, so we will use the Poisson distribution

$$
P_{n}(k)=\lambda^{k} \mathrm{e}^{-\lambda} / k!
$$

Find $\lambda$:

$$
\lambda=n p=100000 \cdot 0.0001=10
$$

The required probability

$$
P_{100000}(5)=10^{5} \cdot \mathrm{e}^{-10} / 5!=10^{5} \cdot 0.000045 / 120=0.0375
$$