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I'm trying to arrive at a formula that calculates the most likely number of people exposed in a group of known size, based on the amount of random exposures in that group.

So lets say there are $100$ people in a group, we randomly select $N$ people, what is the most likely amount of unique people we select? And how do I express that as a function of $N$?

Hope that makes sense.

I'm ideally trying to calculate this in excel, but i can write it in code if need be. I just need a way to calculate this that isn't just writing out a massive table of numbers.

Sebastiano
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JCBoysenBerry
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    Are you selecting $N$ from $100$ without replacement? Are you doing it several times (now with replacement)? The easiest way to find the expected (rather than most likely) number is by linearity of expectation. What have you tried? – Henry Jan 11 '23 at 10:59
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    No replacement. As an example I am giving out 100 dollars 200 times to 100 people. What is the most likely number of people who has received money at least once?

    So same people stay in the group.

    I haven't really tried that much. If you can't tell already i'm not exactly the most experienced in this field, unsure of how to even figure out the answer. So far googling hasn't really helped.

     – JCBoysenBerry Jan 11 '23 at 11:16
  • If you give 100 dollars to 100 people without replacement, then the probability a particular individual gets a dollar is $1$. So that is presumably not what you intended. If you give 10 dollars to 100 people without replacement then the probability a particular individual gets a dollar is $0.1$ and they do not is $0.9$. If you do it twice, the probability they do not get a dollar either time is $0.9^2=0.81$ so the probability they get at least one dollar is $0.19$. And so on. Remember you have 100 individuals – Henry Jan 11 '23 at 11:22
  • Presumably you mean "with replacement" since you want to allow people to be selected more than once, right? – lulu Jan 11 '23 at 11:23
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    Ah yes, sorry i misunderstood. Yes, each person is able to get the money multiple times. My bad. I thought by "replacement" it was meant that people were being replaced after being selected.

    So 100 people in a group. 200 random people will be selected, with each person being able to be selected more than once.

     – JCBoysenBerry Jan 11 '23 at 11:28
  • This is the framework of the coupon collector problem, but you are forcing $N$ coupons to be picked (with replacement), looking at the number of unique coupons you've picked as a distribution (which will be between $1$ and $N$) and you want the mode of this distribution (the most likely number). That seems to be the question, although I could be wrong. – Sarvesh Ravichandran Iyer Jan 11 '23 at 13:24

1 Answers1

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With your example from the comments, of $100$ people and $200$ independent selections (with replacement so people can be selected more than once),

  • the probability a particular individual is selected in a particular selection is $\frac1{100}=0.01$
  • the probability a particular individual is not selected in a particular selection is $1 - \frac1{100} = 0.99$
  • the probability a particular individual is not selected in any of $200$ selections is $\left(1 - \frac1{100}\right)^{200}\approx 0.134$
  • the expected number of individuals not selected in any of $200$ selections is $100\left(1 - \frac1{100}\right)^{200}\approx 13.4$
  • the expected number of individual selected at least once in $200$ selections is $100-100\left(1 - \frac1{100}\right)^{200} \approx 86.6$, and this is easily extended to other cases.

Most likely is harder to calculate than expected. The probability that $m$ different individuals are selected is shown in another question involving Stirling numbers of the second kind. In this case the most likely number is $87$ (close to the expected value) with a probability of about $0.14$, and it is also the median of the distribution.

Henry
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