Question
A fair coin is tossed 12 times. Let A be the event - "exactly 2 tosses are head".
Find the probability distribution of the coin tossing vector $\left(X_{1},\ldots,X_{12}\right) $, given the even A has happened.
My Take
I'm looking to find the probability distribution of the following expression: $$ P\left(X_{1},\ldots,X_{12}=\left(k_{1},\ldots,k_{12}\right)\left|A\right.\right) $$ Which is: $$ \frac{P\left(X_{1},\ldots,X_{12}=\left(k_{1},\ldots,k_{12}\right)\cap A\right)}{P\left(A\right)} $$ I have used binomial distribution to find that: $$ P\left(A\right)={12 \choose 2}\left(\frac{1}{2}\right)^{2}\left(\frac{1}{2}\right)^{10}=66\cdot\left(\frac{1}{2}\right)^{12} $$ but I'm confused with finding $P\left(X_{1},\ldots,X_{12}=\left(k_{1},\ldots,k_{12}\right)\cap A\right)$.
The Book's Answer
The book's answer is $\frac{1}{66}$.
Judging by the book's answer it holds that: $$ P\left(X_{1},\ldots,X_{12}=\left(k_{1},\ldots,k_{12}\right)\cap A\right)=\left(\frac{1}{2}\right)^{12} $$
but I can't tell why.
This probability and $P\left(A\right)$ seems to me like the same one.
in few words... given that "exactly" 2 H occurred, your 12-sized vector is this, for example
$$\{1,1,0,0,0,0,0,0,0,0,0,0\}$$
and all the possible 12-tuple with 10 "zero's" and 2 "one's"
any of these 66 possible tuples are equiprobable with probability
$$\frac{1}{\binom{12}{2}}=\frac{1}{66}$$
If you want you can better formalize this sketch using binomial conditional distribution...
$$\mathbb{P}[\mathbf{X}=\mathbf{x}|A]=\frac{\left(\frac{1}{2}\right)^{12}}{\binom{12}{2}\cdot\left(\frac{1}{2}\right)^{12}}=\frac{1}{\binom{12}{2}}=\frac{1}{66}$$
This because at the numerator you have the probability of one single specified combination you can observe
Examples
$$\mathbb{P}[\{1,1,0,0,0,0,0,0,0,0,0,0\}|A]=\frac{\left(\frac{1}{2}\right)^{12}}{\binom{12}{2}\cdot\left(\frac{1}{2}\right)^{12}}$$
$$\mathbb{P}[\{1,0,0,0,0,0,0,0,1,0,0,0\}|A]=\frac{\left(\frac{1}{2}\right)^{12}}{\binom{12}{2}\cdot\left(\frac{1}{2}\right)^{12}}$$
and so on...
...thus the conditional distribution is uniform all over its support