# Binomial Distribution

I'm trying to solve this question, but since I missed the lecture I'm not where to start, and looking online doesn't help.

Can someone show me how to answer:

Given that X has a binomial distribution with n = 20 and p = 0.39, what is the probability that X takes on the value 8?

I would appreciate all the help I can get.

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You seem to have edited your question to change the problem. The answer given by Michael Hardy isn't really meaningful now. If you want to ask another question, I suggest you write another one. Also, revert the edit so others can understand Michael's answer –  Jean-Sébastien Oct 12 '12 at 0:58

This is one of those instances where the quickest way to find the probability of an event is to find the probability that the event does not happen, and then subtract that from $1$. Thus, find the probability that $X=0$, and subtract that probability from $1$.

It appears that you cannot well afford to skip lectures, since this is one of the most routine problems you'll ever see.

Later note: The question has been changed. It now asks for $\Pr(X=8)$. That's a standard formula. My second paragraph remains intact.

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Thanks for your help, Michael. –  Unknown Oct 12 '12 at 0:42
@Michael Question seems to have been edited, I suggested a roll back, but you might wanna track it to keep your answer in context –  Jean-Sébastien Oct 12 '12 at 1:00

This is if I got your question correctly:

You have 20 identical and independent (i.e. outcomes thereof do not affect each other) experiments, each is either success w.p. $P(S)=0.39$ or failure w.p. $P(F)=1-P(S)$. You do not care, which exactly (out of 20) experiments succeed or fail but you want the probability that exactly 8 thereof succeed.

For a better intuition think of 20 coin tosses w.p. $P(H)=0.39$ and all ways you can get 8 $H$ out of 20 tosses, $\binom{20}{8}$. The probability of this specific outcome is $0.39^8$ and therefore 12 $T \ 0.61^{12}$, hence: $$P(S)=\binom{20}{8} 0.39^8 0.61^{12}$$

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This doesn't match with Michael answers, as you see, the question has been edited to a completly different question. I think we should wait for OP to decide which question he wants to ask –  Jean-Sébastien Oct 12 '12 at 1:01
I agree, but if the OP ants to learn something about Bernoulli experiments, this should be helpful too. –  Alex Oct 12 '12 at 1:03