Statistics probability - finding the exact distribution of X

[Herondaleheir]

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?

Am I even answering this question correctly?

 

[ksdhart2]

The logic seems sound, but what's to stop you from using a variable? Is there some stipulation that says you must manually write out all 100 values of $X$?

$\displaystyle P(X = x) = \binom{100}{x} \cdot \left( \frac{15}{100} \right)^x \cdot \left( \frac{85}{100} \right)^{100-x}$

Side note: You seem to be using $x$ and $X$ interchangeably. Please don't do that. They are two different variables and mean two very different things, especially in stats and probability. There, capital letter variables typically refer to the random variable as a whole, whereas the lowercase variable typically refers to one specific observation of that random variable.