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# Alpha particles are emitted by a radio active source at an average rate of $5$ in a $20$ minutes interval.Using poisson distribution find the probability that there will be at least $2$ emission in a particular $20$ minutes interval .$[e^{-5}=0.0067].$

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!}$$\qquad (x=0,1,2........ for some \lambda > 0) • Constants of a poisson distribution : • Mean=Variance=\lambda • The parameter of the Poisson distribution is \lambda • A Poisson random variable corresponds to rare events. Step 1: Let X be the random variable denoting the number of alpha particles emitted in a 20 minutes interval . X follows a poisson distribution with mean =5 particles in a 20 minutes interval. \therefore \lambda=5 X\sim P(5) P(X=x)=\large\frac{e^{-5}5^x}{x!}$$\quad x=0,1,2.........$
Step 2:
Probability of at least 2 emissions in a 20 minutes interval
$P(X\geq 2)=1-P(X<2)$
$\qquad\qquad=1-(P(X=0)+P(X=1))$
$\qquad\qquad=1-\big(\large\frac{e^{-5}5^0}{0!}+\frac{e^{-5}5^1}{1!}\big)$
$\qquad\qquad=1-0.0067(1+5)$
$\qquad\qquad=1-0.0067(6)$
$\qquad\qquad=1-0.0402$
$\qquad\qquad=0.9598$