# Probability - colored balls in bags

A bag labeled $A$ contains $4$ red balls and $7$ green balls. Another bag $B$ contains $6$ red and $5$ green balls.

A ball is transferred from bag $A$ to bag $B$, after which a ball is drawn from $B$.

Find the probability that it is a red ball?

To be honest I have no idea how to approach the question, I assume that there would be $12$ balls in the bag $B$ when it has been transferred. I'm lost.

• Your assumption is correct. Are you aware of the law of total probability? – GFauxPas Aug 14 '15 at 12:46

Hint:

$$P\left(E\right)=P\left(E\mid R\right)P\left(R\right)+P\left(E\mid G\right)P\left(G\right)$$

Here $R$ is the event that the transferred ball is red, $G$ is the event that the transferred ball is green and $E$ denotes the event that the ball taken out bag $B$ is red.

Second hint: $P\left(E\mid R\right)=\frac7{12}$, why?...

• I understood what you wrote above to be that, E denotes the event that a ball is taken out of bag A, right ? When the ball is drawn from bag B, is it E + Pr(R) from the second bag ? – Mipo Newton Aug 14 '15 at 13:14
• $E$ is (as written) the event that the ball drawn from $B$ is red. If event $R$ has occurred then there are at that moment $7$ red balls in bag $B$ and $12$ in total. That means that $P(E|R)=\frac7{12}$. Here $P(E|R)$ denotes the probability that a red ball is drawn from bag $B$ under the condition that a red ball is transferred from $A$ to $B$. – drhab Aug 14 '15 at 13:19
• Yay! I followed it logically and arrived at the teacher's magical answer.35/66 – Mipo Newton Aug 14 '15 at 14:02

Hint:

Think step by step to come back to cases you know.

1. First step: Transfer a ball

How many different cases are they ? What is the probability of each case ?

1. Second step: for each case of the previous step: draw a ball

What is the probability to draw each ball ?

1. Third step: assemble the results of the previous step

You know the results and the probability of every cases. What are the global results ?

• You should probably label this as a Hint or an Outline – Shailesh Aug 14 '15 at 12:57
• Done, thanks for the advice. – Amaury Aug 14 '15 at 13:01
• Sorry can you explain further ? – Mipo Newton Aug 14 '15 at 13:10
• 1. Transfer a ball from A to B: you either pick a green or red one. You should easily find the probability of each. 2. Let's say you pick a green one. You now know the composition of basket B. You should easily be able to find the probability to pick a green or red ball in basket B. Same process if you pick a red ball from A. 3. You know the probability to pick a green/red ball depending on your first pick. You know the probability to take each ball in A. You should be able to determine the global probability to pick a green/red ball (check drhab's answer). – Amaury Aug 14 '15 at 13:28

You know that ball transferred from A to B is red with 4/11 probability.

Now you have 12 balls in bag B. What is probability it is red? You had 6 red and added 1 with 4/11 probability it is red.

Hint: there are two cases to consider. One case is that a green ball is moved to bag $B$. The other case is a red ball is moved to bag $B$.

• Yes, I got from @drhab, P(E)=P(E∣R)P(R)+P(E∣G)P(G) – Mipo Newton Aug 14 '15 at 13:16
• I can visualize it, not just mathematically. – Mipo Newton Aug 14 '15 at 13:17