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).