probability

[datonman]

Hello, i was hoping someone can help me understand this:

Tracy has 3 bags of marbles, each bag containing at least 1 blue marble, at least 1 red marble, and no marbles of another color. If Tracy selects 1 marble at random from each bag, what is the probability that all 3 marbles that she selects will be red?

(1)There is a total of 5 red marbles and 5 blue marbles in the 3 bags.
(2)The ratios of red to blue marbles in the 3 bags are 2:1, 1:1, and 1:2

[MartyMurray]

Tracy has 3 bags of marbles, each bag containing at least 1 blue marble, at least 1 red marble, and no marbles of another color. If Tracy selects 1 marble at random from each bag, what is the probability that all 3 marbles that she selects will be red?

So we are seeking to determine if given the information in the statements there is only one possible probability that all three marbles are red.

(1)There is a total of 5 red marbles and 5 blue marbles in the 3 bags.

I have a feeling that there will be multiple probabilities possible given this constraint. So I am going to see if by distributing the marbles in two different ways I can get different probabilities, in which case this statement will be insufficient.

We need at least one of each in each bag. So with start with RB RB RB.

Let's distribute the rest so that we get RRRB RB RBBB.

The probabilities of drawing a red from each bag are respectively 3/4, 1/2, and 1/4. The probability of drawing three reds is 3/4 x 1/2 x 1/4 = 3/32.

Ok now let's distribute the marbles in a way that is clearly different from the first way. RRB RRB RBBB.

The probabilities for drawing a red from each bag are 2/3, 2/3 and 1/4.

The probability of getting three reds is 2/3 x 2/3 x 1/4 = 2/36, which is clearly different from 3/32.

So given the constraint described in Statement 1 we can create two different scenarios that generate two different probabilities of choosing 3 reds. So Statement 1 is insufficient.

(2)The ratios of red to blue marbles in the 3 bags are 2:1, 1:1, and 1:2

This is interesting. We don't know how many marbles are in each bag, but we know that we are drawing one from each. So what really matters are the three probabilities of drawing a red marble from each bag. We can then use those three probabilities to calculate the probability of drawing three red marbles.

If the ratios of red marbles to blue marbles in the three bags are 2:1, 1:1, and 1:2, then the respective probabilities of drawing a red marble from each are 2/3, 1/2 and 1/3. From those we can calculate the probability of choosing three red marbles. We don't need the answer, just to know that we can get it.

So Statement 2 is sufficient and the correct answer is B.