A worker tends three looms that work independently of each…

A worker tends three looms that work independently of each other. The probability that a loom will not require the worker's attention during one hour is $0.9$ for the first, $0.8$ for the second and $0.7$ for the third. Find the probability that during one hour at least one loom will not require the worker's attention.

Step 1: Events

$A_k$ = "loom $k$ needs no attention", $P(A_1)=0.9$, $P(A_2)=0.8$, $P(A_3)=0.7$, independent.

Step 2: The opposite event

All three need attention:

\[ P(\overline{A_1}\,\overline{A_2}\,\overline{A_3})=0.1\cdot0.2\cdot0.3=0.006. \]

Step 3: The answer

\[ \boxed{P(A_1+A_2+A_3)=1-0.006=0.994.} \]

Step 4: Other questions about the same looms