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Suppose I had two groups of people $A$ and $B$. I want to know the probability that the next person a randomly sampled person from group $A$ interacts with comes from group $B$.

Obviously the population distribution of group $A$ and $B$ matter, that is $P(A)$ and $P(B)$. But there are also population dynamics (say segregation or homophily) that increase the likelihood that people in group $A$ interact with people in group $A$ and people in group $B$ interact with people in group $B$.

What is the probability exactly one of the next five people a person from $A$ meets will be in group $B$? What is the probability it is exactly two people. Is there a distribution that describes this?

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This is much harder if the populations are small and the probability is not constant.

However, if the populations are fairly large, say $70\%$ adults and $30\%$ children and all people are equally likely to interact with one another, we have the probability that the next $5$ interactions by an adult is a child would be $(30\%)^5=0.243\%$.

When we have segregation and homophily/heterophily involved, this becomes slightly more convoluted, and we may have to stratify groups (e.g. adult $\rightarrow $ adult male & adult female and proceed with simple probability from there).

Sometimes, we have bias, e.g. if an adult was twice as likely to interact with another adult as another child (homophily), we would ACTUALLY have an $82\%$ chance of an adult-adult interaction and an $18\%$ chance of an adult-child reaction.

Only the binomial distribution works for simple probabilities like the first example I gave. All other scenarios can be decomposed into binomial scenarios. If it happens to be a larger scenario, like the probability that an adult interacts with $30$ adults in his next $50$ interactions, a normal distribution can accurately estimate the probability with computational ease.

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  • $\begingroup$ Thank you, this is most helpful. What is the math behind your 82/18 calculation? $\endgroup$ – Beth Feb 20 '18 at 15:09
  • $\begingroup$ If adults are twice as likely to interact with another adult, instead of 70/30 we get 140/30. Obviously the probabilities have to add up to 100, so we scale it down to 82/18. $\endgroup$ – Saketh Malyala Feb 20 '18 at 19:13

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