Can Binomial Distribution Be Used for Small Populations?

[QuickLoris]

Hello all!
I'm trying to understand whether I can use the binomial distribution in a certain way...

According to the equation, to find the probability P of a certain number of successes out of a number you trials, you need the number of trials, n; the number of successes out of the trials, x; and the probability of a success on any given trial, p.

Now let's say you have a small population size, N = 40, of LEDs, that either work or don't. Can I assume p to be .50 and and set P to .95 so I can determine what n and x I would need? Or am I supposed to do a preliminary study to determine p?

Are there other tests I should do instead for finding the probability of failure in small population and small sample sizes? Is there a minimum % of the population that I should test?

I haven't found any problem like this in any of the textbooks I've looked through.

 

[DeShark]

I'm not quite sure what you're trying to achieve here. Do you want to know how many LEDs will be defective on average?

Just because you only have two outcomes doesn't mean your probability is 0.5 - every time you cross the road, you can die or live. This doesn't mean you have a 50% chance of death each time you cross. You will need to do a study to determine the p-value, along witha confidence interval. The bigger your sample size, the smaller your confidence interval will be. As I don't know what your intention is, I can't give a better answer at the moment.

 

[cdux]

A Binomial problem (an extension of a Bernoulli one) is quite clear in dealing with the determination of probability to have a certain number of hits based on a certain number of tries and a particular probability of success of each try.

The rest you mention about determining the actual probability of success of a particular hit from a sample population is another subject entirely.

In case you're wondering if the Binomial distribution works for a small number of tries, given that you know the probability of success of each try, it's perfectly safe. In fact, it's not an approximation and directly a result of basic probability axioms (i.e. it's accurate even for 1 or 2 tries). It's when it gets to approximations such as the Central Limit Theorem via the N(), that large samples might be required.

 

[QuickLoris]

Thank you for your replies. I think I understand now. The binomial distribution is a predictive measure of the probability of getting a certain amount of successes given a specific number of trials and a defined probability of success for each independent trial. So, it's not meant to be used to test a hypothesis.

 

[DeShark]

It could be used to test a hypothesis. The null hypothesis would be : The probability of failure is < 5%. If you take a sample of 50 parts and get 5 failures, the probability of this is 6.6% so you would not reject the null hypothesis at the 95% confidence level. I.e. 5 defective parts is not enough to go back to the LED manufacturer and complain.