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.

 

[blue_raver22]

Your understanding is partially correct. The binomial distribution with probability k for success has an average of Nk. This means that if you were to repeat the experiment many times, on average, you would expect to have Nk successes. However, the distribution is not necessarily peaked around this average value. It can be spread out depending on the value of N and k.

To answer your question, it is not accurate to say that the probability of having one or more successes in N tries is simply Nk. This only holds if Nk is less than or equal to 1. If Nk is greater than 1, then the probability of having at least one success is actually 1. This is because having one or more successes is guaranteed if Nk is greater than 1.

In general, the probability of having at least one success in N tries can be calculated using the binomial distribution formula: P(≥1) = 1 - (1-k)^N. This formula takes into account the fact that there can be more than one success in N tries.

To determine when the binomial distribution is a good approximation for the probability of having one or more successes, you can look at the shape of the distribution. If the distribution is relatively narrow and peaked around the average value of Nk, then it is a good approximation. However, if the distribution is spread out and not peaked around Nk, then it may not be a good approximation. This is dependent on the values of N and k, so there is no specific cutoff for when it is a good approximation. It is always best to use the formula for calculating the probability of at least one success to get an accurate result.