there is a ladder with 4 bars, a ground bar marked as bar 0. no bars beneath bar 0. no bar above bar 4. A robot starts climbing it up from bar 0. He will climb up or down the bars depending on the probability, at each bar, there is different probability to climb up as follows:
to climb up successfully:
from bar 0 to bar 1 : 1/3
bar 1 to bar 2: 1/4
bar 2 to bar 3: 1/5
bar 3 to bar 4: 1/6
if he fails, he will only drop down 1 bar. ex: from bar 3 to bar 2.
Question: what is the chance for robot climbing up to the 4th bar within N times?