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.
 A: Hint:
Think step by step to come back to cases you know.


*

*First step: Transfer a ball


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


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


What is the probability to draw each ball ?


*Third step: assemble the results of the previous step


You know the results and the probability of every cases. What are the global results ?
A: 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?...
A: 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.
A: 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 $B$.
