I've been trying to figure out the following question:
I modelled the discrete random variable X with a binomial distribution such that:
X ~ Bin(3, 3/5)
I took the number of trials to be 3 because there can be between zero and three faulty buses (successful trials) on the Monday in question.
I took the probability of having one faulty bus on a given day to be 3/5. I calculated this by taking the probability of one bus on one day to be 1/5 (because there are five days) and multiplying by three (because there are three faulty buses made that week). This gave me Pr(success) = 3/5
When I used the equation V(X) = npq (n=trials, p=success probability, q=failure probability) I found the variance of X to be equal to 18/25. (npq = 3 x 3/5 x 2/5 = 18/25)
In the answers for this question though, the variance is listed as being 12/25:
I don't know what I've done wrong here. Have I missed out some fundamental bit of information? Is the answer sheet wrong?
Any help is massively appreciated. Thank you for taking the time to read all of this :)
There are $3$ independent experiments, each of them represented by a faulty bus, and with equal probability to succeed.
We speak of a success if a faulty bus is made on Monday.
Then we are dealing with binomial distribution with parameters $n=3$ and $p=\frac15$.
That gives variance $np(1-p)=3\frac15\frac45=\frac{12}{25}$