Example 6.6. Two machines work independently of each other. The probability of the first machine operating without failure over a certain period of time $t$ is $p_{1}=0.9$, and for the second machine $-p_{2}=0.8$. What is the probability of both machines operating without failure over the specified period of time
Solution. Consider the following events: $A_{1}$ and $A_{2}$ - trouble-free operation of the first and second machines, respectively, over time $t$; $A$ - trouble-free operation of both machines over the specified time. Then the event $A$ is the conjunction of events $A_{1}$ and $A_{2}$, i.e., $A=A_{1} A_{2}$. Since events $A_{1}$ and $A_{2}$ are independent (the machines operate independently of each other), we obtain by formula (5):

$$
P(A)=P\left(A_{1}\right) \cdot P\left(A_{2}\right)=0.9 \cdot 0.8=0.72
$$