A meeting has 12 employees. Given that 8 of the employees are female, what is the probability that all employees are female? 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.
 A: 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.
A: 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.
A: 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?
A: 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".
A: 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.
