Problem:
An "unfair" coin has a 2/3 probability of turning up heads. If this coin is tossed 50 times, what is the probability that the total number of heads is even?
Answer Choices:
A. 25(32​)50
B. 21​(1−3501​)
C. 21​
D. 21​(1+3501​)
E. 32​
Solution:
Let p be the probability that the total number of heads is even, and let q be the probability that the total number of heads is odd. Since the probability of tossing k heads and (50−k) tails is (k50​)(32​)k(31​)50−k, we have
p=(050​)(32​)0(31​)50+(250​)(32​)2(31​)48+⋯+(5050​)(32​)50(31​)0
and
q=(150​)(32​)1(31​)49+(350​)(32​)3(31​)47+⋯+(4950​)(32​)49(31​)1
Note that p−q=(32​−31​)50=3501​. Since p+q=1, we solve for p to get p=21​(1+3501​).