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