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.