Problem:
When a certain biased coin is flipped times, the probability of getting heads exactly once is not equal to and is the same as that of getting heads exactly twice. Let , in lowest terms, be the probability that the coin comes up heads exactly times out of . Find .
Solution:
Let be the probability of getting heads when the coin is tossed once. Then the probability of getting heads out of tosses is . Consequently, the information given in the problem leads to the equation
from which or . Since the desired probability is not , we may conclude that . Thus the probability of obtaining exactly heads in tosses is
and .
The problems on this page are the property of the MAA's American Mathematics Competitions