Answer
a) $0.155$
b) $8\times 10^{-5}$
c)$6.93\times 10^{-4}$
d) $10.8$
Work Step by Step
Let $X$ be the random variable of number of reactions. $X$ has the binomial distribution with parameters $n=20, p$, We can calculate the parameter $p$ directly from the table given in the exercise $3-32:$
$$
p=\frac{48+60}{200}=0.54
$$
The probability mass function of $X$ is given by:
$$
\mathbb{P}(X=x)=\left(\begin{array}{l}
20 \\
x
\end{array}\right) 0.54^{x} \times(0.46)^{20-x}, x=0,1, \ldots, 20
$$
Calculate using this formula:
$$
\begin{array}{l}
\mathbb{P}(X=12)=0.155 \\
\mathbb{P}(X \geq 19)=\mathbb{P}(X=19)+\mathbb{P}(X=20)=0.00008=8\times 10^{-5}\\
\mathbb{P}(X \geq 18)=\mathbb{P}(X=18)+\mathbb{P}(X=19)+\mathbb{P}(X=20)=6.93\times 10^{-4}\\
\mathbb{E}(X )=np=20\times 0.54=10.8
\end{array}
$$
