Answer
a) $0.246$
b) $0.055$
c) $0.01$
d) $0.322$
Work Step by Step
$X$ is binomial random variable with the parameters: $n=10, p=0.5 .$ Calculate:
(a)
$$ \mathbb{P}(X=5) =\left(\begin{array}{c}{10} \\ {5}\end{array}\right) 0.5^{10}=0.246$$
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(b)
$$ \mathbb{P}(X \leq 2) =\mathbb{P}(X=0)+\mathbb{P}(X=1)+\mathbb{P}(X=2)=\\ =\left(\begin{array}{c}{10} \\ {0}\end{array}\right) 0.5^{10}+\left(\begin{array}{c}{10} \\ {1}\end{array}\right) 0.5^{10}+\left(\begin{array}{c}{10} \\ {2}\end{array}\right) 0.5^{10}=0.055
$$
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(c)
$$\mathbb{P}(X \geq 9)= \mathbb{P}(X=9)+\mathbb{P}(X=10)=\\=\left(\begin{array}{c}{10} \\ {9}\end{array}\right) 0.5^{10}+\left(\begin{array}{c}{10} \\ {10}\end{array}\right) 0.5^{10}=0.01 $$
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(d)
$$\mathbb{P}(3 \leq X\lt5)=\mathbb{P}(X=3)+\mathbb{P}(X=4)=\\=\left(\begin{array}{c}{10} \\ {3}\end{array}\right) 0.5^{10}+\left(\begin{array}{c}{10} \\ {4}\end{array}\right) 0.5^{10}=0.322 $$
