I am running across two conceptual inconsistencies with the formula for binomial distributions (Binomial distribution - Wikipedia, the free encyclopedia):

$\displaystyle \binom{n}{k}P[success]^k (1-P[success])^{(n-k)}$

These are problems I've come across when trying to apply this formula, and I don't know if the reasons for them are mistakes on my part or something I'm not understanding about when the formula is applicable. Help!

The first problem is in the following example. Assume that in a sample of air, there is Pk probability of there being a certain type of molecule. It follows that in n units of air if I apply a test to find that molecule (assume that the test always succeeds if the molecule exists, always fails otherwise) I have a Pk chance of finding it. But if I take that same sample of air and break it up into n samples of 1 unit of air each, and apply the same test to each sample, the probability that I will find the molecule in any of those samples is:

$\displaystyle \binom{n}{1}P_k^1 (1-P_k)^1 = nP_k(1-P_k) $\neq$ P_k$

it's easy to find values of Pk and n for which the inequality holds. How can this be right? Doesn't it mean that given a certain Pk and n, the chance of finding the molecule in the very same sample of air increases based on how we measure it?

The second problem is one I came across while working on this problem: http://www.mathhelpforum.com/math-he...tml#post614608. Basically if there is a (possibly unequal) distribution for a 6-sided die, and the probability of getting a 2 is P2, and if two die are rolled and you are told at least one die is 2, what is the probability the sum of the two die is 4?

The "normal" method of figuring this out is described in the post I linked above, which yields the answer $\displaystyle \frac{P_2}{1-P_2}$. But treating it as a bernoulli trial yields:

$\displaystyle \binom{2}{1}P_2^1(1-P_2)^1 = 2P_2(1-P_2)$

which is clearly different. What is going on here?