181. a) The device consists of a large number of independently operating elements with the same (very small) probability of failure of each element over time $T$. Find the average number of elements that fail over time $T$, if the probability that at least one element will fail during this time is 0.98.
Solution. From the condition of the problem (since the number of elements is large, the elements work independently, and the probability of failure of each element is small), it follows that the number of failures is distributed according to the Poisson law, and it is required to find the parameter $\lambda$ (the average number of failures).

The probability that at least one element will fail, according to the condition, is 0.98, hence (see problem 179, part g), $1-\mathrm{e}^{-\lambda}=0.98$. From this,

$$
\mathrm{e}^{-\lambda}=1-0.98=0.02
$$

Using the table of the function $\mathrm{e}^{-x}$, we find $\lambda=3.9$. Thus, during the time $T$ of the device's operation, approximately four elements will fail.

b) Find the average number $\lambda$ of defective items in a batch of items, if the probability that this batch contains at least one defective item is 0.95. It is assumed that the number of defective items in the considered batch is distributed according to the Poisson law.

Hint. Take $\mathrm{e}^{-3}=0.05$.