1557. Electric lamps are manufactured at 3 plants. The first plant produces $45 \%$ of the total number of lamps, the second - 40, the third - 15. The production of the first plant contains $70 \%$ standard lamps, the second - 80, the third - 81. Products from all three plants are supplied to stores. What is the probability that a lamp purchased in a store will be standard?
Solution. Let's introduce the following notations. Let event $A_{1}$ be that the purchased lamp is manufactured at the first factory, $A_{2}$ - the lamp is from the second factory, $A_{3}$ - the lamp is from the third factory, and finally, event $B$ - the lamp turns out to be standard. From the problem statement, it follows that $P\left(A_{1}\right)=0.45, P\left(A_{2}\right)=0.40, P\left(A_{3}\right)=0.15 ; P\left(B / A_{1}\right)=$ $=0.70, \quad P\left(B / A_{2}\right)=0.80, P\left(B / A_{3}\right)=0.81$. Since $B=A_{1} B+$ $+A_{2} B+A_{3} B$, then $P(B)=P\left(A_{1}\right) P\left(B / A_{1}\right)+P\left(A_{2}\right) \quad P\left(B / A_{2}\right)+$ $+P\left(A_{3}\right) P\left(B / A_{3}\right)=0.45 \cdot 0.70+0.40 \cdot 0.80+0.15 \cdot 0.81=0.7565$.