Problem:
An unfair coin lands on heads with a probability of 41​. When tossed n times, the probability of exactly two heads is the same as the probability of exactly three heads. What is the value of n ?
Answer Choices:
A. 5
B. 8
C. 10
D. 11
E. 13
Solution:
The probability of exactly two heads is (n2​)(41​)2(43​)n−2, and this must equal the probability of three heads, (n3​)(41​)3(43​)n−3. This results in the equation
2n(n−1)​⋅43​=6n(n−1)(n−2)​⋅41​ or 83​=24n−2​
Therefore n=11​.
The problems on this page are the property of the MAA's American Mathematics Competitions