Q: Scientists collecting butterfly samples near a nuclear power plant found that wing and antenna defects were somewhat common. Of ninety-eight butterflies collected in the sampling, three showed at least one defect. If a previous study on butterflies found near the same nuclear power plant thirty years found that there was one defect per thousand butterfly, what is the probability of finding seven in a sample of ninety-eight?
- So I found the probability of finding a defect in present time: 3/98 = 0.0306
- Then I found the probability of the study 30 years ago: 1/1000 = 0.001
- I ignored the 3/98 statistic at the beginning b/c I think it's irrelevant
- Then I found the probability of observing 7/98 which is 0.07143
- I then multiplied 0.07143 and 0.001 together and got 0.007143% as my answer
I'm confused as to how a binomial distribution can be used in this problem... unless my answer is incorrect.
You have $1/1000$ as the probability that any particular butterfly will have a defect.
You are told that for the sample of ninety eight, at least three do have the defect, and wish to know the probability that exactly seven do when given that condition.
That is, find $\mathsf P(X=7\mid X\geq 3)$ when $X\sim\mathcal {Bin}(98, 1/1000)$.