Question

A random variable X has the following probability distribution:

X :	1	2	3	4	5	6	7	8
P (X) :	0.15	0.23	0.12	0.10	0.20	0.08	0.07	0.05

For the events E = {X : X is a prime number}, F = {X : X < 4}, the probability P (E ∪ F) is

(a) 0.50
(b) 0.77
(c) 0.35
(d) 0.87

Answer 1

(b) 0.77

E = {X is a prime number} = {2, 3, 5, 7}

$$\begin{gathered} P\left(E\right)=P\left(2\right)+P\left(3\right)+P\left(5\right)+P\left(7\right)=0.62 \\ F=\{X<4\}=\{X=1,2,3\} \\ P\left(F\right)=P\left(1\right)+P\left(2\right)+P\left(3\right)=0.5 \\ \mathrm{Now}, \\ E\cap F=\{2,3\} \\ P\left(E\cap F\right)=P\left(2\right)+P\left(3\right)=0.35 \\ P\left(E\cup F\right)=P\left(E\right)+P\left(F\right)-P\left(E\cap F\right)=0.62+0.50-0.35=0.77 \end{gathered}$$