Find the mistake in my conditional probability calculation

Question

A meeting has $12$ employees. Given that $8$ of the employees are women, find the probability that all the employees are women?

I just defined the two events $A : 8$ employees are female and $B :$ all employees are female. Thus, we need $P(B|A)=\frac{P(A \cap B)}{P(A)}$.

Now, we know that $P(A \cap B) = P(B)$. So, $P(B)=\frac{1}{2^{12}}$ considering equal probability of male and females. Also, $P(A)={12 \choose 8} \frac{1}{2^{12}}$. On dividing the conditional probability comes out to be $\frac{1}{{12 \choose 8}}$ but it does not match with the answer to this question. Where am I going wrong? Anyone please help!

Answer 1

Your calculation of $P(A)$ gives you the probability that exactly $8$ employees are female, not the probability that at least $8$ employees are female. To calculate this, take $$ \left({12 \choose 8}+{12\choose 9}+...+{12 \choose 12}\right)2^{-12}. $$

Answer 2

The probability of a female employee is not the same as that of a male employee.

If the probability is $2/9$, then the probability of a female employee can be found by solving for $p$:

$$\frac 29=\frac{p^{12}}{\sum_{i=8}^{12}{12\choose i}p^i(1-p)^{12-i}}$$

Thus $p\approx.88153$