Answer
$P(a~particular~fingerprint)=\frac{1}{2^{32}}$
Work Step by Step
For simplicity, consider:
Event A = "all 24 squares filled in correcty".
Event B = "determining the fingerprint type".
Event C = "correct number of ridges"
$P(all~24~squares~filled~in~correcty)=P(A)=\frac{1}{2^{24}}$
$P(determining~the~fingerprint~type)=P(B)=(\frac{1}{2})^4=\frac{1}{2^4}$
$P(correct~number~of~ridges)=P(C)=(\frac{1}{2})^8=\frac{1}{2^8}$
Now, using the Multiplication Rule (see page 282):
$P(a~particular~fingerprint)=P(A~and~B~and~C)=P(A)\times P(B)\times P(C)=\frac{1}{2^{24}}\times \frac{1}{2^4}\times \frac{1}{2^8}=\frac{1}{2^{32}}$
