Answer
a) $2.4 \times 10^{-8}$
b) $0.9999$
c) $9.91 \times 10^{-18}$
d) $1.14 \times 10^{-4}$
Work Step by Step
$X$ is binomial random variable with the parameters: $n=10, p=0.01$
Calculate:
(a)
$$
\mathbb{P}(X=5)=\left(\begin{array}{c}{10} \\ {5}\end{array}\right) 0.01^{5} 0.99^{5}=\left[2.4 \times 10^{-8}\right]
$$
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(b)
$\begin{aligned} \mathbb{P}(X \leq 2) &=\mathbb{P}(X=0)+\mathbb{P}(X=1)+\mathbb{P}(X=2)=\\ &=\left(\begin{array}{c}{10} \\ {0}\end{array}\right) 0.01^{0} 0.99^{10}+\left(\begin{array}{c}{10} \\ {1}\end{array}\right) 0.01^{1} 0.99^{9}+\left(\begin{array}{c}{10} \\ {2}\end{array}\right) 0.01^{2} 0.99^{8}=0.9999 \end{aligned}$
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(c)
$\begin{aligned} \mathbb{P}(X \geq 9) &=\mathbb{P}(X=9)+\mathbb{P}(X=10)=\\ &=\left(\begin{array}{c}{10} \\ {9}\end{array}\right) 0.01^{9} 0.99^{1}+\left(\begin{array}{c}{10} \\ {10}\end{array}\right) 0.01^{10} 0.99^{0}=9.91 \times 10^{-18} \end{aligned}$
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(d)
$\begin{aligned} \mathbb{P}(3 \leq X<5) &=\mathbb{P}(X=3)+\mathbb{P}(X=4)=\\ &=\left(\begin{array}{c}{10} \\ {3}\end{array}\right) 0.01^{3} 0.99^{7}+\left(\begin{array}{c}{10} \\ {4}\end{array}\right) 0.01^{4} 0.99^{6}=1.14 \times 10^{-4} \end{aligned}$
