<s>[INST] Conditional Probabilities multi coloured balls from a bag I have two bags
Bag1 : Contains 4 red and 5 black balls
Bag2 : Contains 3 red and 7 black balls
One ball is drawn from bag1 and 2 balls are drawn form bag2. Find the probability of 2black balls and 1 red ball.
How do I go about solving this problem?.
This is what I have come up with so far: 
$$P(B,RR) + P(R,BB) = \frac{5}{9} \cdot \frac{7}{10} \cdot \frac{3}{9} + \frac{4}{9} \cdot \frac{7}{10} \cdot \frac{6}{9}$$
Thanks.
 [/INST] 
This is what I have come up with so far:
$$P(B,RR) + P(R,BB) = \frac{5}{9} \cdot \frac{7}{10} \cdot \frac{3}{9} + \frac{4}{9} \cdot \frac{7}{10} \cdot \frac{6}{9}$$

That's not quite okay.   The probability of drawing the red from bag-2 is twice that.   It is the old "two ways to draw the same thing" problem: red then black, black then red.
$$P(B,RR) + P(R,BB) = \frac{5}{9} \cdot \frac{7}{10} \cdot \frac{3}{9} + \frac{5}{9} \cdot \frac{3}{10} \cdot \frac{7}{9} + \frac{4}{9} \cdot \frac{7}{10} \cdot \frac{6}{9}$$
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