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# $20\%$ 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.$[e^{-2}=0.1353].$

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
$P(X=2)=\large\frac{e^{-2}2^2}{2!}$
$\qquad\qquad=2\times 0.1353$
$\qquad\qquad=0.2706$