A machine operates if all of its three components function. The p… - Vidyadip

MCQ

A machine operates if all of its three components function. The probability that the first component fails during the year is 0.14 , the second component fails is 0.10 and the third component fails is 0.05 . What is the probability that the machine will fail during the year?

A
0.1542
✓
0.2647
C
0.3642
D
0.4231

✓

Answer

Correct option: B.

0.2647

(b) : Consider the following events:
$A=$ First component of the machine fails during the year
$B=$ Second component of the machine fails during the year
$C=$ Third component of the machine fails during the year
We have, $P(A)=0.14, P(B)=0.10$ and $P(C)=0.05$
Clearly, the machine will fail if at least one of its three components fails during the year.
$
\begin{array}{l}
\text { Required probability }= P (A \cup B \cup C) \\
=1-P(\overline{A \cup B \cup C})=1-P(\bar{A} \cap \bar{B} \cap \bar{C}) \\
=1-P(\bar{A}) P(\bar{B}) P(\bar{C})[\because A, B, C \text { are independent events }] \\
=1-(1-0.14)(1-0.10)(1-0.05) \\
=1-(0.86)(0.90)(0.95)=0.2647 .
\end{array}
$

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