<s>[INST] Distribution of amount of green balls drawn Urn of type A contains 4 green and 3 blue balls. Urn of type B contains 4 blue and 5 green balls.
There are three urns of type A and two urns of type B. We pick one urn at random (out of 5) and we draw 3
balls (with a single draw) out of the chosen urn. Let X denote the number of green balls drawn. Determine
the distribution of X.
I tried the following approach
$
P(X=N)=\frac{3}{5}\binom{3}{N}\frac{\binom{4}{N}\binom{3}{3-N}}{\binom{7}{3}}+\frac{2}{5}\binom{3}{N}\frac{\binom{5}{N}\binom{4}{3-N}}{\binom{7}{3}}
$
where $N \in \{ 0,1,2,3 \}$
However sum of probabilities is greater than 1:
$
\sum_{N=0}^{3} \frac{3}{5}\binom{3}{N}\frac{\binom{4}{N}\binom{3}{3-N}}{\binom{7}{3}}+\frac{2}{5}\binom{3}{N}\frac{\binom{5}{N}\binom{4}{3-N}}{\binom{7}{3}} = \frac{283}{105}
$
I have a problem with understanding how to calculate the probability of obtaining N green balls from a single draw.
 [/INST] To better understand the question I think it would be better to consider a random variable $Y$ to the problem. Let $Y$ be the random variable that indicates which urn was chosen.  Note that
$$
\begin{array}{ll}
\mathbb{P}(Y=A)=3/5 
&
\mathbb{P}(Y=B)=2/5 
\\
\mathbb{P}(X=n | Y=A)=\frac{\binom{4}{n}\binom{3}{3-n}}{\binom{7}{3}} 
&
\mathbb{P}(X=n | Y=B)=\frac{\binom{5}{n}\binom{4}{3-n}}{\binom{9}{3}}
\end{array}
$$
Now just use the following principle to calculate the required probability.
\begin{align}
\mathbb{P}\Big(X=N\Big) & =\mathbb{P}\Big(X=n| Y=A \Big)\mathbb{P}(Y=A) + \mathbb{P}\Big( X=N|Y=B \Big)\mathbb{P}(Y=B)
\end{align}
</s>