Bag A contains 1 red and 2 white marbles, and Bag B contains 1 white and 2 red marbles. A marble is randomly chosen from Bag A and placed in Bag B. A marble is then randomly chosen from Bag B. Determine the probability that the marble selected from Bag B is white.

I'm having some trouble with these "marbles-in-bags" questions, I'd really appreciate if someone could explain this one to me. Thanks! ^^

Question

Bag A contains 1 red and 2 white marbles, and Bag B contains 1 white and 2 red marbles. A marble is randomly chosen from Bag A and placed in Bag B. A marble is then randomly chosen from Bag B. Determine the probability that the marble selected from Bag B is white.

I'm having some trouble with these "marbles-in-bags" questions, I'd really appreciate if someone could explain this one to me. Thanks! ^^

kropot72 · Answer

P(drawing white ex bag A) = 2/3
P(drawing white ex bag B) =
$$\frac{1+\frac{2}{3}}{4}$$

nikolas · Answer

@kropot72 thanks for your answer but I'm still a little confused. I understand that the 4 in the denominator is there because now there are 4 balls in the second bag - but where does the 1 in the numerator come from?

anonymous · Answer

Make a tree diagram.  The answer is 5/12.  I'm uploading the tree diagram now.

anonymous · Answer

Had to redraw it the original sketch was pure chicken scratch.

anonymous · Answer

attachment

anonymous · Answer

Hopefully that makes sense.  The ball you draw from Bag B is dependent on the ball you draw first from Bag A. 
 If you draw a Red from Bag A first then there are now 4 balls in Bag B to choose from, but 3 of them are red.
If you draw a White from Bag A first then there are still now 4 balls in Bag B to choose from, but now it's even with 2 being Red and 2 being White.

anonymous · Answer

All this problem is concerned with though is the probability of drawing a White ball from Bag B. So you only focus on the branches with that outcome. (I put stars next to them in the sketch)

anonymous · Answer

The rule is you multiply along connected lines.  
And then add those products of the probabilities that you want.

anonymous · Answer

I can go over the theory without the tree diagram, if you'd like, but it's really wordy.  Tree diagrams when dealing with bernoulli like events are much more succinct and accessible.

anonymous · Answer

Anyway, I hope that helped. Take care and don't forget to select a Best Response. Regards.

nikolas · Answer

That's excellent, thank you very much! What helped the most was what you said about the first set of branches only dealing with the probability of drawing a white ball from the first bag (2/3), I kept on missing that for some reason.

anonymous · Answer

It's cool and you're welcome.  I'm a big fan of probability and statistics...but starting out, it kinda kicked my butt a bit.  Take care and best of luck with everything going forward.