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If I use Bayes' theorem here , event A denoting that 12 employees are female and event B denoting that 8 employees are female, (assuming that an employee has equal chances of being a male or female) I get,

$P(A \mid B) = \frac{P(B \mid A) \times P(A)}{P(B)} = \frac{1 \times (0.5)^{12}}{\binom{12}{8}(0.5)^{12}}=1/\binom{12}{8}$

Is this the correct way of doing this? I am especially confused because the gender of the employees are independent of one another , and yet I am using the information that 8 of them are female to determine the probability that all of them are female.

I am sorry if I sound confused and do not make sense.

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    $\begingroup$ if I know they are females, can't I just calculate the probability of the remaining 4 employees to be females? 0.5^4 $\endgroup$
    – rep_ho
    Jun 13, 2021 at 21:12
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    $\begingroup$ Exactly where my confusion is. This makes sense , and yet I cannot figure out where in my calculation using Bayes' did I do a mistake. Thats the only use of the information that us provided. $\endgroup$
    – noobcoder
    Jun 13, 2021 at 21:15
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    $\begingroup$ can you use | isntead of / where appropriate? and also write explicitly what do you think p(A) is, and p(B) is $\endgroup$
    – rep_ho
    Jun 13, 2021 at 21:25
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    $\begingroup$ $12 \choose 8$ is the number of ways for exactly 8 employees to be female, not for at least 8 employees to be female. You presumably want the latter, since the probability that all 12 are female given that exactly 8 are female is just 0. $\endgroup$
    – fblundun
    Jun 13, 2021 at 21:29
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    $\begingroup$ I'm saying it's impossible that the number of female employees is simultaneously equal to exactly 8 and exactly 12. $\endgroup$
    – fblundun
    Jun 14, 2021 at 8:05

5 Answers 5

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The confusion comes from the fact that there are multiple ways to interpret "Given that 8 employees are female":

  • If it's 8 specific employees - say, the employees in positions 1 thru 8 - then the remaining four have $2^4$ possible gender configurations, only $1$ of which is all-female, giving $\frac{1}{2^4}$

  • If it's any 8 of the 12 employees, then what's being asked is to look at all configurations of 12 employees, throw out the ones with 5 or more men, and count the proportion that are all female.

    Notice that under this interpretation, each employee in the valid configurations does not have a 50% chance of being male/female, since we are assuming that there are at least 8 females in each valid configuration. What does have an equal chance is each valid configuration.

The reason this is confusing is that our intuition assumes the first interpretation, but the way the question is worded implies the second.


There is a famous statistical "paradox" that stems from this same line of reasoning:

In a family with two children, one of whom is a girl, what's the probability both are girls?

Most people assume the answer is $\frac{1}{2}$, but it's actually $\frac{1}{3}$, for the same reason as the original question. If you're still confused, see this answer which gives a more thorough explanation of the paradox and its resolution.

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    $\begingroup$ "I'll give you \$100 if you tell me truthfully there is at least one girl, and I'll give you \$10,000 if you tell me truthfully there are two girls". Answer: "At least one girl". Probability of two girls is zero. $\endgroup$
    – gnasher729
    Jun 14, 2021 at 9:49
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    $\begingroup$ I think your paradox example is wrong. I see it as four scenarios, boy girl (mentioned), girl (mentioned) boy, girl (mentioned) girl, girl girl (mentioned). Two out of four scenarios have two girls. $\endgroup$
    – Tvde1
    Jun 14, 2021 at 12:26
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    $\begingroup$ To make it even more complicated, depending on the wording, the probability can depend on how you know the information. $\endgroup$ Jun 14, 2021 at 17:15
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    $\begingroup$ The main part of this answer correctly explains that the wording of the question was ambiguous and didn't contain enough information to correctly calculate the desired probability. But then you cite 'a famous statistical "paradox"' which contains the same kind of ambiguity, and you conclude "Most people assume the answer is 1/2, but it's actually 1/3."? No it's not. The only possible answer is "can you please rephrase the question, it does not contain enough information, I can't give you a probability with these words". $\endgroup$
    – Stef
    Jun 15, 2021 at 11:58
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    $\begingroup$ I strongly disagree with the statement "The reason this is confusing is that our intuition assumes the first interpretation, but the way the question is worded implies the second.". Someone tells you they know that 8 of the 12 employees are female. How did they get that information? Did they know personally 8 of the 12 employees (first interpretation), or did they use a computer program to build a list of potential 12-employee groups, then throw away the groups that did not have at least 8 female employees (second interpretation)? Occam's razor would suggest the first interpretation. $\endgroup$
    – Stef
    Jun 15, 2021 at 12:04
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Perhaps it would be helpful to give this some clearer structure, via explicit assumptions. Suppose we are willing to assume a priori that each person is equally likely to be male or female, and we assume that the sexes are mutually independent. Then the "female-indicator" variables for the people in the group are:

$$X_1,...,X_{12} \sim \text{IID Bern}(\tfrac{1}{2}).$$

Consequently, the number of females in the group has a binomial distribution:

$$\dot{X} \equiv \sum_{i} X_i \sim \text{Bin}(12, \tfrac{1}{2}),$$

and the conditional probability of interest is:

$$\mathbb{P}(\dot{X} = 12 | \dot{X} \geqslant 8) = \frac{\mathbb{P}(\dot{X} = 12)}{\mathbb{P}(\dot{X} \geqslant 8)} = \cdots$$

Can you take it from here?

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    $\begingroup$ Hello Ben, another question from a beginner: Is this approach fully Bayesian? I tried to answer the question by assuming a Beta(2,2) prior for the Bernoulli parameter $\mu$ (the person being a female), then maybe generate Bernoulli samples (with size 12) given the posterior $\mu|\text{Data}$ and calculate the probability of all persons being females. $\endgroup$
    – Algo
    Jun 14, 2021 at 6:07
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    $\begingroup$ You can certainly do that if you wish --- if you are unwilling to assume equal probability of males and females then you would give that probability parameter a prior and then proceed via Bayes theorem. $\endgroup$
    – Ben
    Jun 14, 2021 at 8:58
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You need to be very, very, very precise with the statements you make, otherwise any results will be utter nonsense - because they might be the correct answer to a totally different questions.

My reading of your question as asked leads to the answer "the probability is zero". Eight out of twelve employees are female, so four are male, so not all employees are female.

Let's interpret it as "Someone picked 12 employees at random, and counted how many were female. The answer was a number from eight to twelve". Or "Someone picked 12 employees at random, then picked eight of those and checked their gender, and all eight were female". Much different situation, and much different answer.

In the first case, if it was nine females, why did I say "the answer was a number from eight to twelve" and not "the answer was a number from nine to twelve"? If it was eight females, why didn't I say "the answer was a number from zero to eight"? I might have an agenda to make the impression that either lots of employees or that few employees are female, so if you don't know about the agenda, you might get different answers.

Let's say I ask you "how many children do you have" and you answer "two". Then I say "I very much prefer boys to girls. So if you tell me that you have at least one boy, I'll give you 100 dollars. If you tell me that you have two boys, I'll give you 10,000 dollars. If you lie, I'll shoot you". If you tell me "I have at least one boy" then the probability that you have two boys is zero.

But still, you can't answer the question at all. We don't know how many employees there are - because your question wasn't clear. I know there were 12 employees in a meeting, but I don't know how many were outside the meeting. Obviously the more outside the meeting, the less likely it is that they are all female. And we don't know the probability that a random employee is female. You guessed that the probability was 0.5. I would assume that the probability is an unknown number, that the chances that eight employees are female depends on that number, and vice versa you can draw conclusions from the number of females in a group what the probability is that some employee is female.

So let's restate the question. You picked 12 employees at random. I tell you "I will ask you about some attribute that each employee might or might not have, and I want you to tell me if the number of employees in the group with that attribute is from eight to twelve or not".

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  • $\begingroup$ We know even less. Most meetings don't have random employees attending (and I cannot find that word in the original statement). If you have meeting at a hospital it could for instance be a meeting for doctors or nurses - with different probabilities. Finally the assumption that all attendees are iid is questionable in practice. $\endgroup$ Jun 15, 2021 at 7:20
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    $\begingroup$ If we are going to interpret "8 of the employees are female" as "at least 8 of them", shouldn't we interpret "a meeting has 12 employees" as "a meeting has at least 12 employees"? $\endgroup$
    – Abigail
    Jun 15, 2021 at 13:28
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What you're noticing here is that the entire field of statistics is plagued by serious interpretational issues.

Chief among these (and at fault here) is the reference class problem. In a frequentist framework, this corresponds to assigning your statement of "probability" to a well-populated space of outcomes (recall that probability may be axiomatized in terms of event spaces - in a Bayesian framework, the formalization is slightly different, but the practical consequences are the same). For a statement such as the one in this problem, there are multiple ways we could conceivably do this - what's worse, there's no obvious "correct" statistical interpretation purely from the wording of the problem. We need more information.

This is one of the main problems with statistics as it is currently taught. Mathematical pedagogy tends to favor short, snappy exercises - they're easy to read and easy to grade. But statistics abhors this; any sufficiently short statistical statement is almost bound to be uninterpretable.

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Is not there also a potential skewing of the probabilities due to "cultural" (for want of a better word) factors. If 8 of the employees are female, perhaps this is a Women's gym that does not hire men, or perhaps it is a small company run by an entrepreneur who (probably unlawfully, but that is another subject) only or primarliy hires women? Perhaps this meeting is of members of an occupation that skews female like nursing or elementary school teaching.

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    $\begingroup$ this is a question about a contradiction that arises in probability theory when events or sets aren't correctly defined. it's not a question about societal demographics or culture. $\endgroup$ Jun 15, 2021 at 20:48

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