Engineer here. Math wasn't so tough personally but I got absolutely destroyed and mauled in economics. I wasn't expecting Macro to be very theoretical. I too hope the economics cutoff is 60ish or less, I may get in then.
About the monkey question, it can be solved by using binomial distribution and summing probabilities which form a geometric series. Alternatively you can put values of k, and look for the answer that makes sense. Couple of them were absurd and could be ruled out at sight.
Probability it reaches k= Probability it reaches in 1jump, plus probability it reaches in two jumps and so on. Individually these are going to be bionomial.
P is fixed, it can take any value between half and one, but we begin our analysis by fixing P. If p keeps varying jump to jump as you hypothesised it would be an interesting question but a near impossible solution for me.