3.2.5 problem 20

(a)

$P(X=k)=\left(\frac{n}{k}\right)p^k{(1-p)}^{n-k}$ for $0\le k\le 3$.

(b)

To use the complement of the desired deven, $$P(X>0)=1-P(X=0)=1-{(1-p)}^3=1-(-p^3+3p^2-3p+1)=p^3-3p^2+3p.$$

To prove the same by Inclusion-Exclusion,

$$P(X>0)=\sum _{i=1}^{3}P(I_{X_i}=1)-3p^2+P({\cap }_{i=1}^3{I}_{X_i}=1)=3p-3p^2+p^3.$$

(c)

Since $p^2$ and $p^3$ go to $0$ asymptotically faster than $p$, when $p$ is small, $3p-3p^2+p^3\approx 3p$.