So I drew it out on the back of a napkin. There is one possible path to P and T.

There are four possible paths to Q and S, as you showed.

There are six possible paths to R.

The probability distribution is:

1 - P

4 - Q

6 - R

4 - S

1 - T

I bet a nice normal distribution arises if you let go of enough balls, with a bulge at R.

That got me thinking that the probability of Q is then 4 out of 16. "And that's our answer."

But, instinctively, I thought:

"If it hits the first peg/point, then it's a 50-50 shot at going either way. The problem is at the second two pegs. There, it's 3/4 that it can go into Q, and the 1/4 that it goes to RST, and 1/4 that it goes to P. At the next pegs, it's 50-50 that it can go to Q, but only depending on which way it went at the previous pegs." So, with prior knowledge, I have to adjust my odds. Same with 4. And Bayesian statistics kicked into high gear in my head, and that's how I got to 0.25.

That's the biostatistician in me: Gets the right answer, but is meaningless without more information.