These are numeric solutions when it was too complicated or long winded to solve the problem by hand.
Problem 1b: Probability of choosing two pairs of socks both red is 1/2 and an even number of black pairs
→ python3 p1b.py
...
r + b = N: 15 + 6 = 21, P = 1 / 2.0
Probability that gambler will be behind after 36 rolls in about 3/4
>>> 2*(37/38)**35*(36.5/38)
0.7553943362209891