# Homework Help: Probability: Red ball being transfefred from bag m to n?

1. May 24, 2012

### mirandab17

Bag M contains 5 white balls and 2 red balls. Bag N contains 3 white balls and 4 red balls.
a) A ball is randomly selected from Bag M and Placed in Bag N. A ball is then
randomly selected from Bag N. What is the probability that the ball selected form
Bag N is white?
b) If a white ball is selected from Bag N, what is the probability that a red ball was
transferred from Bag M to Bag N?

a) I got this answer which is right: 13/28. I got this by taking the probablity that a white ball is chosen from bag m and then that a white ball is chosen from bag n (which now has one more white ball) and then multiplying that number by the probability of choosing a red ball from bag m, but then choosing a white ball which is the same, just one more red ball to choose from.

That sounds horribly complicated.
Overall: (5/7)(4/8) + (2/7)(3/8) = 13/28

b) This is where I get confused... I know it's a conditional probability problem, so GIVEN THAT the white ball is selected from bag n (my previous answer), aka (13/28) must be on the bottom... but how do I get what's on the top? A red ball transferred from Bag M to N... so I thought it must be (2/7)(5/8)...

2. May 24, 2012

### Ray Vickson

Let W1 = first is white, R1 = first is red, W2 = second is white. You have computed P(W2) = P(W2|W1)P(W1) + P(W2|R1)P(R1). You want P(R1|W2). This is P(R1 & W2)/P(W2), and we can write P(R1 & W2) = P(W2 & R1) = P(W2|R1) P(R1). You know P(W2|R1), P(R1) and P(W2), so you have everything you need.

RGV

3. May 25, 2012

### mirandab17

Okay... I don't understand how P(R1&W2) turned into P(W2/R1) P (R1). Why are you multiplying it?

4. May 25, 2012

### Villyer

Here you are saying that a red ball was picked from bag M and a red ball from bag N.
But the question states that a white ball was picked from bag N.

5. May 25, 2012

### mirandab17

But it is conditional right? So IF a white ball was picked from bag N, which is 11/21, the probability of transferring a red ball from M to N is... i'm assuming red was picked first here then... so (2/7) and then white (3/8)...?

6. May 25, 2012

### mirandab17

so (2/7)(3/8) / (11/21)?

7. May 25, 2012

### Villyer

I think it's conditional. I don't know anything about probability outside of what I can deduce logically, but from what I've seen this matches that definition.

And your numerator is right now, but double check the fraction you placed in the denominator.

8. May 25, 2012

### mirandab17

The denominator should be (3/7) because it's only involving bag N?

9. May 25, 2012

### Villyer

You're original denominator was correct.