Could you use the binomial distribution here?

1. Mar 8, 2013

trollcast

I'm looking through my statistics notes and on the page thats giving examples of cases where you can use a binomial distribution it gives the problem:

"The number of red counters in a randomly chosen sample of 30 counters taken from a large number of counters of which 10% are red."

Now my notes goes on to say that this can't be modelled by a binomial distribution but doesn't say how you could model it with any other distribution.

But given that very limited amount of data could you not obtain a reasonable estimate of the probabilities using the binomial distribution as the question states, "a large number" , could we not assume that removing the counter isn't going to change the probability very much?

2. Mar 8, 2013

MrAnchovy

Yes, the binomial distribtion is appropriate here. I think your notes should say that "if the number of counters is not large it cannot be modelled using the binomial distribtion"; in this case you would have to calculate the specific probabilities of 0, 1, 2... red counters when selecting 30 from N (without replacement - if there is replacement then the binomial distribution always applies).

3. Mar 8, 2013

trollcast

Thanks,

How would you define large enough? If the sample is 1% of the population or something?

4. Mar 8, 2013

MrAnchovy

Firstly, I should have pointed out that "calculating the specific probabilities of 0, 1, 2... red counters when selecting 30 from N without replacement" is in fact the Hypergeometric Distribution.

"Large enough" depends on how accurate you want to be; for further investigation and limits on errors see statistical text books or google "binomial hypergeometric difference".

5. Mar 8, 2013

trollcast

Ok, I thought that if population was too small then it wouldn't work at all but I see now how its all to do with a small population making the error far too large to get a sensible value out of it.

6. Mar 8, 2013

"Large enough" depends on how accurate you want to be; for further investigation and limits on errors see statistical text books or google "binomial hypergeometric difference"."

Typically if the sample size is at most 5% of the population size the binomial distribution can be used.

7. Mar 8, 2013

MrAnchovy

Indeed. In this case the binomial distribution gives P(3) ≈ 0.24 whereas with N = 60, P(3) ≈ 0.33.

8. Mar 10, 2013

Redbelly98

Staff Emeritus
The population size must be large so that the 10% value is true for the entire collection of 30 counters -- the binomial distribution assumes the same probability (10% in this example) for all 30 counters to be picked.

If you sample without replacement from a small population, then removing a counter significantly affects the likelihood that the next counter chosen is red. But if you sample with replacement, then the 10% value remains fixed no matter the population size -- as MrAnchovy indicated, the binomial distribution would apply exactly, not just approximately.