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I've N=1000 balls and X=20 bins. I distribute them randomly in these 20 bins. Let's say I've y=100 balls in one of the bins (let's say bin y). What is the probability to have y=100 balls in this bin. Can I use a binomial as :

pbinom(100,1000,1/20,lower.tail=F)
3.908209e-11

And can I compute the p-value for each bin using the same method?

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    $\begingroup$ Can you word your question more clearly? You say you have 100 balls in one bin then ask what the probability is to have 100 balls in that bin. By the premise, the probability is 1. $\endgroup$
    – AdamO
    Commented Jan 19, 2017 at 14:27
  • $\begingroup$ 100 balls out of 1000 balls. I distribute these 1000 balls randomly in 20 bins. $\endgroup$
    – Nicolas Rosewick
    Commented Jan 19, 2017 at 14:31
  • $\begingroup$ To be completely explicit - by "randomly" do you mean for each ball every bin has an equal chance to get the ball? (it might well be that you intend that the bin be selected at random but with unequal probabilities, for example) $\endgroup$
    – Glen_b
    Commented Jan 20, 2017 at 3:56

1 Answer 1

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Whether assignment probabilities are equal or unequal, the joint probability for all bins will follow a multinomial distribution, and therefore the marginal distribution for any single bin will be binomial. As such, you should use the dbinom function instead (unless you, in fact, want $P(|y|\leq 100)$).

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  • $\begingroup$ Could you clarify that initial phrase about "equal or unequal"? It seems to me that some forms of unequal assignment probabilities will produce decidedly non-multinomial distributions, depending on what you mean by "assignment probability." $\endgroup$
    – whuber ♦
    Commented Jun 7, 2018 at 13:19
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    $\begingroup$ If you assume a generating model of selecting one bin (with replacement) and placing a single ball into said bin, and repeat, the distribution is clearly multinomial. This is probably what people intuitively think when they say random assignment. Now there are certainly other assignment models (pick a Poisson number of bins, assign based on Dirichlet probabilities, etc.) that are not multinomial, but will require changing the bin selection framework significantly, as any sequential random selection (any discrete distribution over boxes) with replacement will produce a multinomial distribution. $\endgroup$
    – deasmhumnha
    Commented Jun 7, 2018 at 17:16
  • $\begingroup$ Of course, the selection events must be i.i.d. as well for the joint to be multinomial. $\endgroup$
    – deasmhumnha
    Commented Jun 7, 2018 at 17:23

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