Binomial Distribution: Average & Probability of ≥1 Success

[aaaa202]

The average if the binomial distribution with probability k for succes is simply:

<> = Nk

So this means that if <> = 1 the distribution function must be peaked around 1. In general when is it a good approximation (i.e. when is the function peaked sufficiently narrow) to say that the probability in N tries to have one or more succeses is simply:

P(≥1) = Nk

this obviously does not hold for Nk>1 but on the other hand I don't expect it to hold for small N. So my guess is when Nk is sufficiently small. Is that correct?

 

[mathman]

P(at least one success) = 1 - P(all failure) = 1 - (1-k)^N.
You should be able to analyze it.

 

[aaaa202]

Right so using the binomail theorem you find:

P(at least one success) = Nk - K(N,2)k² + K(N,3)k³ - K(N,4)k⁴ + ...

So the question is when the first term dominates. I am guessing for sufficiently small k?

 

[mathman]

You need Nk small, not just k. In your expression you seem to have extraneous K (capital k) from the second term on.