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

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$X$ denote the number of bad cheques received by the bank in a day with average number of such cheques being 6$6$ i.e.E(X)=6$\mathbb{E}(X)=6$. We can use Poisson distribution to model the uncertainty in such cases i.e. X∼Pois(6)$X\sim \text{Pois}(6)$. So the probability that X$X$ is equal to 4$4$ is

Pr(X=4)=64e−64!${\displaystyle Pr(X=4)=\frac{6^4{e}^{-6}}{4!}}$

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