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Q)

$20\%$ of the bolts produced in a factory are found to be defective. Find the probability that in a sample of $10$ bolts chosen at random exactly $2$ will be defective using ,Poisson distribution.$ [e^{-2}=0.1353].$

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A)
Toolbox:
  • A random variable $X$ is said to have a poisson distribution if the probability mass function of $X$ is
  • $P(X=x)=\large\frac{e^{\Large -\lambda }\lambda^x}{x!}$$\qquad (x=0,1,2........$ for some $\lambda > 0)$
  • Constants of a poisson distribution :
  • Mean=Variance=$\lambda$
  • The parameter of the Poisson distribution is $\lambda$
  • A Poisson random variable corresponds to rare events.
Step 1:
Let $X$ be random variable denoting the no of defective bolts in a sample of 10 bolts chosen at random.
Probability that a bolt is defective=0.2
$n=10$
$\therefore \lambda=np$
$\Rightarrow 10\times .2=2$
Step 2:
Using a Poisson distribution :
$X\sim P(2)$
$\Rightarrow P(X=x)=\large\frac{e^{-2}2^x}{x!}$$\qquad x=0,1,2......$
Step 3:
Probability of exactly 2 defective bolts
$P(X=2)=\large\frac{e^{-2}2^2}{2!}$
$\qquad\qquad=2\times 0.1353$
$\qquad\qquad=0.2706$
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