Probability that it was the second coin that showed that showed head? 
Two coins are flipped. One of the coins is such that it shows heads
  with probability $1/3$ whereas the second one shows heads with
  probability $2/3$. Given that you get exactly one heads, what is the
  conditional probability that it was the second coin?

I really don't get how to intuitively think through this. I want to use the formula for conditional probability
$$P(A|B)=\frac{P(A\cap B)}{P(B)},$$
But I don't know how to find $P(A\cap B)$. Anyone with a good explanation on how to master problems like these?
 A: Let $X_i = 1$ if coin $i$ is heads, with $i=1,2$. So we want to calculate $$\mathbf P(X_2 =1\,| \, X_1 + X_2 = 1).$$
We use Bayes Rule in the form
$$ \mathbf P(A \, | \, B)  =  \frac{\mathbf P(B \, | \, A) \mathbf P(A)}{\mathbf P(B)},$$
which gives us
\begin{align*}
\mathbf P(X_2 =1\,| \, X_1 + X_2 = 1) & =
 \frac{\mathbf P( X_1 + X_2 = 1\, |\,X_2 =1) \, \mathbf P(X_2= 1)}{ \mathbf P(X_1 + X_2 = 1)}\\
& =  \frac{\mathbf P(  X_1 =0) \, \mathbf P(X_2= 1)}{ \mathbf P(X_1 + X_2 = 1)}\\
& =  \frac{\mathbf P(  X_1 =0) \, \mathbf P(X_2= 1)}{ \mathbf P(  X_1 =0) \, \mathbf P(X_2= 1) + \mathbf P(  X_1 =1) \, \mathbf P(X_2= 0)}\\
& = \frac{(2/3 \times 2/3) }{(2/3 \times 2/3) +  (1/3 \times 1/3)}\\
& = \frac{4}{5},
\end{align*}
A: Let:


*

*$A_1$ - first coin shows heads ($P(A_1)=\frac{1}{3}$)

*$A_2$ - second coin shows heads ($P(A_2)=\frac{2}{3}$) 


Of course situations $A_1$ and $A_2$ are independent.


*

*$B_1$ - only first coin shows heads ($P(B_1)=P(A_1)P(A_2')=\frac{1}{9}$)

*$B_2$ - only second coin shows heads ($P(B_2)=P(A_1')P(A_2)=\frac{4}{9}$)

*$B$ - only one coin shows heads (as far, as $B_1\cap B_2=\emptyset$, we have $P(B)=P(B_1)+P(B_2)=\frac{5}{9}$


Notice, that $B_2=A_2\cap B$
Now we have
$P(A_2|B)=\frac{P(A_2\cap B)}{P(B)}=\frac{P(B_2)}{P(B)}=\frac{4}{5}$
A: Hints:
1) The sample space: $$H_1H_2;H_1T_2;T_1H_2;T_1T_2.$$
2) The required probability is the ratio of probability of "the second heads outcome" to the sum of probabilities of "single heads outcomes".
