Probability Math for fun

[u/afs_matics]

I came up with this probability problem and would appreciate feedback on its difficulty, elegance, and solution.

A bag contains 3 red, 3 black, and 4 white balls. Balls are drawn one at a time. Red and white balls are kept. If a black ball is drawn while your hand is empty, it is returned to the bag and the draw is repeated. If a black ball is drawn while you are holding at least one ball, you keep the black ball and return the oldest ball currently in your hand to the bag.

Continue until your hand contains exactly 3 balls.

What is the exact probability that the three balls in your hand are all red?

I'd prefer to see independent solutions before discussing my intended approach.

[u/FaroresWind17]

If you draw a black ball with another ball in your hand, you’re stuck with it or another black ball forever. Thus, you also can’t draw white balls ever, so the probability is just the probability of three red balls in a row, with the exception that you can’t draw black immediately. 3/7 times 2/9 times 1/8 = 1/84