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I understand for a random variable to be distributed binomially:

  • There must be n fixed trials
  • Each trial must be independent of the others
  • Probability of success must remain constant throughout the trials
  • Outcome either happens or doesn't happen

So take this set up (sorry for the big chunk of text):

A university scholarship committee must select two students to receive a certain scholarship. The committee receives eight applications for the scholarships - five from male students and three from female students. The applicants are all equally qualified, so the two scholarship winners are randomly selected from the eight applicants. Let X be the number of female students who receive a scholarship.

From what I understand, X is indeed a binomial random variable but this is where my confusion arises.

Is X ~ Bin(3,3/8) or X ~ (8,3/8)?

I wasn't sure if n refers to all 8 possible students or just the 3 female students.

kiwizor
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  • The trials are not independent. Knowing that, say, the first male student was accepted reduces the chances that the first female student was accepted. – lulu Feb 10 '23 at 12:14
  • If you like, you can imagine that each student is distinguished, not just by gender (so we have $M_1, \cdots, M_5, F_1, F_2, F_3$. Then, since any pair is equally likely to be drawn, you are just making a single selection from a set with $\binom 82$ elements. – lulu Feb 10 '23 at 12:15
  • @Lulu oh so, say I were to pick a female student for the first scholarship, then that means the probability of picking the female student the next one is going to be 2/7. So the probability isn't fixed and so X isn't binomially distributed? – kiwizor Feb 10 '23 at 12:18
  • You could draw a simple tree diagram to find the probabilities of $0, 1$ or $2$ females – David Quinn Feb 10 '23 at 12:47
  • @DavidQuinn yea I understand how to work out the probabilities, just less so the definition of X as a binomial random variable – kiwizor Feb 10 '23 at 12:52
  • It isn't binomial, because of the dependence between the trials. It's still easy to count, of course. – lulu Feb 10 '23 at 13:00
  • If you know how to work out the probabilities, then you can see the values are not from a Binomial Distribution. I hope it’s clear why it’s not Binomial. – David Quinn Feb 10 '23 at 13:30
  • @DavidQuinn yes I see that the probability isn't constant throughout the trials thanks – kiwizor Feb 10 '23 at 19:20

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