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Probability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsProbability1 Mark
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.
$0.2647$
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.
Required probability $= P (A \cup B \cup C)$
$=1-P(\overline{A \cup B \cup C})$
$=1-P(\bar{A} \cap \bar{B} \cap \bar{C})$
$=1-P(\bar{A}) P(\bar{B}) P(\bar{C})[\because A, B, C$ are independent events $]$
$=1-(1-0.14)(1-0.10)(1-0.05)$
$=1-(0.86)(0.90)(0.95)=0.2647 .$

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