Consider the binomial tree in Section 1.2.2. Suppose that the probabilities of “moving up and down” on the first day equal 1/2, but on the second day, the price moves up with a probability of p₁ if on the first day it moved up, and with a probability of p₂ if on the first day it dropped.

(a) Find the probability that on the second day the price will equal ud.

(b) Given that the price took on the value ud, find the probability that it moved up on the first day.