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# Perform The following a) If a bank receives on an average λ=6 bad cheques per day what is the probability

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Perform The following
a)  If a bank receives on an average λ=6 bad cheques per day what is the  probability that it will receive 4 bad cheques on any given day.

b)  A farmer buys a quantity of cabbage seeds from a company that  claims that approximately 90% of the seeds will germinate if planted  properly. If four seeds are planted, what is the probability that  exactly two will germinate.

A)

Let $X$ denote the number of bad cheques received by the bank in a day with average number of such cheques being $6$ i.e.$\mathbb{E}\left(X\right)=6$. We can use Poisson distribution to model the uncertainty in such cases i.e. $X\sim \text{Pois}\left(6\right)$. So the probability that $X$ is equal to $4$ is
$Pr\left(X=4\right)=\frac{{6}^{4}{e}^{-6}}{4!}$
To get this value, one can use R:

x = 4
lambda = 6
dpois(x, lambda)



and the value is:

Pr(X=4)=0.1338526

B)

This situation follows the binomial distribution with n=4 and p=90/100=9/10
The random variable X is the number of seeds that germinate. We have to calculate the probability that exactly two of the four seedswill germinate. That is P[X=2]. By applying Binomial formula , we get

P[X=2] = (4C2) * (9/10)2 *(1/10)2
= 6 * (81/100) * (1/100) = 486/10000 = 0.0486
So , the required probability is 0.0486