104. In the observations of Rutherford and Geiger, a radioactive substance emitted on average $3.87 \alpha$-particles over a period of 7.5 seconds. Find the probability that this substance will not emit any $\alpha$-particles in one second.
Solution. The random variable $X$ - the number of $\alpha$-particles emitted by a substance in one second, is distributed according to the Poisson law. The mathematical expectation of the variable $X$ is $\lambda=3.87 / 7.5=0.516$. The required probability that $X=0$ is $P=e^{-\lambda}=e^{-0.515}=0.5669$.