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I have the population which is split into three classes; A, B and C. We know that the number of observations in each class is Na, Nb and Nc respectively. Assuming that my observations are binary distributed how can I specify (with a formula) probability of observing not more than P positive outcomes.

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What makes an observation positive for each class? I assume you mean by "binary distributed" that in each class there is some probability of having a positive outcome, and the outcome is thus bernoulli distributed? Is there some (perhaps relevant?) reason for the strange naming of the number of observations for each class? Apart from these weirdnesses, this feels quite a bit like a homework assignment... If so: please tag it as such, and take the time to type in the assignment properly. – Nick Sabbe Oct 19 '11 at 13:39
Hi. Yes, I'm talking about bernoulli distributed values. It's not a homework. The naming is wrong. typo. I'll fix it. Thanks – user333 Oct 19 '11 at 14:17
I'll repeat @Nick's questions, because you replied but did not address them. Precisely what constitutes a "positive" outcome? Indeed, what is an "observation"? Here are two possible different interpretations: (1) you sample a population and observe which of three classes each member of the sample belongs to, yielding three class counts. (2) You stratify a population into three classes and observe some binary attribute ("positive"/"negative"), giving three counts of "positive" outcomes. – whuber♦ Oct 19 '11 at 14:39
I don't have any empirical data at hand.let's Assume that we have 30 class A observations (read this patients) 60 class B patients and 200 class C patients. If I denote Pa to be probability of observing positive outcome for class A (patient died) and follow the notation for B and C I could say that probability that not a single patient died in group A is Pa^Na... I was wondering how can I express (in terms of ps and ns what is the probability of having not more than 5 patients who died... Hope this better explains the problem – user333 Oct 19 '11 at 15:52
"...positive outcome for class A (patient died) ..." The patient must have been in really bad shape if death was a positive outcome! But I don't understand where probability is coming in. Are you choosing a patient at random from class A and determining if the patient is alive? – Dilip Sarwate Oct 19 '11 at 16:06

1 Answer

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You have three classes of observation $A$, $B$ and $C$, with $N_A$, $N_B$ and $N_C$ observations respectively. The probability of a particular observation being positive is $p_A$, $p_B$ and $p_C$ depending on the class. You want to know the probability of no more than $k$ positive outcomes (using $P$ might be confusing).

This is $$\sum_{a=0}^k \sum_{b=0}^{k-a} \sum_{c=0}^{k-a-b} {N_A \choose a}{N_B \choose b}{N_C \choose c} p_A^a p_B^b p_C^c (1-p_A)^{N_A-a} (1-p_B)^{N_B-b} (1-p_C)^{N_C-c}.$$

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Looks like what I need. Would you mind providing little bit of explanations – user333 Oct 19 '11 at 17:10
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You want the probability of $a$ positive from $A$, $b$ positive from $B$ and $c$ positive from $C$ where $a+b+c \le k$. So you use the binomial distribution three times and do a lot of adding up. – Henry Oct 19 '11 at 18:31
Very good. I'll look up binomial distribution formulae on Wikipedia - hope your expression will make more sense after that. – user333 Oct 19 '11 at 20:28

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