[Math] Statistics probability – finding the exact distribution of X

probabilityprobability distributionsstatistics

So I'm stuck on finding the most efficient way to approach this homework problem.

In a certain population, 15% of the people have Rh-negative blood. A blood bank serving this population receives 100 blood donors on a particular day.

a. Let X be the number of donors in the sample with Rh-negative blood. What is the
exact distribution of X?

So I'm assuming in order to solve this question, I'm supposed to do something like:

P(x=0) = ( 100 C 0 )*(15/100)^0 (85/100)^100

P(x = 1) = (100 C 1) (15/100)^1(85/100)^99

P(x = 2) = (100 C 2)*(15/100)^2 (85/100)^(98)

….

But that seems like an awful lot of steps for one question. Is there a quicker way to answer this question wihout using combinations? Should I not be using binomial distribution?

Update: After doing some searching I found a thread with a similar question. Someone commented using poisson distribution and I was wondering if that would work? Exact and approximate probability distribution

Best Answer

Comment in Answer format to permit a graph.

Assuming that participation does not change with blood type, the number with Rh negative blood will be $X \sim \mathsf{Binom}(n = 100, p=0.15).$

In R statistical software a binomial PDF is called dbinom, so the formula in @ sasquire' comment is as shown in the code below, used to make the figure.

x = 0:30;  PDF = dbinom(x, 100, .15)
plot(x, PDF, type="h", lwd=2)
abline(h = 0, col="green2")

enter image description here

Because $P(X > 30) \approx 0,$ that part of the graph is not shown.

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