Is there a faster way to do this than laying out the whole list? Thank you.

Jennifer, actually laying out the whole list is the fastest way for doing this, in my opinion. You can try and see how long it takes. In my case when I did it, it took less than a minute.

The probability of picking 3 from the jar is 1/4, without replacement the probability of picking zero on your second grab becomes 1/3, 1/4*1/3 = 1/12 for (3,0), repeat,

(0,3) = 1/4*1/3 = 1/12,

The other possible combinations is (2,1) and (1,2) each with a probability of 1/12,

1/12 + 1/12 + 1/12 + 1/12 = 4/12, reduce = 1/3

I did it a lot faster. I thought, okay, there are two combinations that give a sum of 3: 1+2 and 0+3.

So, I took each one separately. For 1 and 2: it doesn't matter which one I draw first. So the probability of drawing a 1 or a 2 is 2/4=1/2. Then there are 3 marbles left, and assuming I have one correct number, there is only one other correct number left, so the probability is 1/3. Multiply those and you have 1/6.

I started to do the same for 0 and 3 but quickly realized that it was the same patterns what I just did, just different numbers. So that gives another 1/6.

1/6 + 1/6 = 2/6 = 1/3.

Why do you have to repeat the inversion? As in (3,0) and (0,3). If I pull out a 3 and then a 0, that's the same thing in the long run as pulling out a 0 and a 3, given that they add up to 3. There are really only two distinct ways of adding up to 3: (1,2) or (2,1), and (0,3) or (3,0) --as in, since we're adding the terms, the commutative property of addition should apply and should be that the order doesn't matter. That's why I'm arguing that there are only two unique ways to add up to 3. Therefore, it's 1/4*1/3= 1/12.

Can you clarify why pulling out a zero the. A 3 is different than pulling out a 3 then a zero?

And what happen with (0,0)?

I thought of it this way: After I pull the first marble, there are 3 marbles left. No matter which number I pull first, there is always exactly 1 marble, no more no less, that will lead me to add up to 3. So the answer is 1/3.