I am learning probability and found an exercise I could not get:
An investment analyst has tracked a certain blue-chip stock for the past six months and found that on any given day it either goes up a point or down a point. Furthermore, it went up on 25% of the days and down on $75\%.$
What is the probability that at the close of trading four days from now the price of the stock will be the same as it is today? Assume that the daily fluctuations are independent events.
The answer is $0.211,$ but I just tried to order the pieces, and found that based on the six months that is $180$ days and based on the $25\%$ of days going up and $75\%$ going down, that is $45$ and $135$ days.
So if $n$ is $180$ and $k$ is $45$ and $135,$ what is $p?$
How can I operate: ${n \choose k} p^k (1-p)^{n-k}.$ Thanks a lot!