Problem:
An unfair coin has probability p of coming up heads on a single toss. Let w be the probability that, in 5 independent tosses of this coin, heads come up exactly 3 times. If w=625144​, then
Answer Choices:
A. p must be 52​
B. p must be 53​
C. p must be greater than 53​
D. p is not uniquely determined
E. there is no value of p for which w=625144​
Solution:
We must solve for p when
(35​)p3(1−p)2=10p3(1−p)2=625144​,
or p3(1−p)2=72/55. If we define
f(p)=p3(1−p)2−5572​
we see that f(0)=f(1)=−72/55, and
f(21​)=321​−5572​≈.03−.024>0
Thus, since f is continuous, f has at least two real roots, r1​,r2​, satisfying 0<r1​<21​ and 21​<r2​<1. In fact, r1​=2/5 and 3/5<r2​<4/5.