A box contains 5 red and 5 green balls. A ball is selected at random and its color is noted.

Question

If this ball is red, it is replaced; if it is green, it is not replaced. Then, a second ball is drawn and its color is noted. Draw a tree and find the probability that:

1. (a) the second ball is red.
2. (b) the second ball is red given that the first was green.

I know that in total there are 10 balls in the box so that would be 5/10 for both R (red) and G (green) on the first tree branch. After this, I get a bit confused. I'm not sure if it stays 5/10 throughout the entire tree or the numbers change.

Thank you for the help!

Answer 1

A key piece to remember is what happens to the tree based on the first selection, and to remember that the overall probability is affected by the first choice. So for (a)...

(a) Since there are equal numbers of red and green on the first choice, the probability is 5/10 or 1/2 or 0.5 that you would pick either on the first roll. From there, though, things change since in one case you replace the ball (red) and the other, you don't (green). So you need to allow for that in the calculations as follows:

Probability of red, if red on the 1st: (1/2) * (1/2) = (1/4) or 0.25

Probability of red, if green on the 1st: (1/2) * (5/9) = 5/18 or 0.28

Overall probability: 0.25 + 0.28 = 0.53 or 53 percent (which makes sense since you've removed the green if you roll it on the first roll).

(b) In the second example, you already know you've rolled a green. As such, you don't need to consider that as part of the final probability. Given that, you know that the green stays out and the probability of getting red on the next pick is 5/9 or 0.56.

Answer 2

The 2nd ball being red could happen as a result of two events:

select red, then select red

probability of selecting red is 5/10, ball is replaced so probability of selecting red again is 5/10, so probability of selecting red, then red is 5/10 * 5/10 = 1/4

select green, then select red

probability of select green is 5/10, ball is not replaced, so probability of selected red is 5/9, so probability of selecting green then red is 5/10 * 5/9 = 5/18

sum probabilities to determine probability of selecting red as second ball: 1/4 + 5/18\\

2nd ball being red, given 1st ball is green

the ball was not replaced, leaving 4 green balls and 5 red balls, so the probability 5/9.