Consider the Venn diagram in the next column, where

$\begin{array}{l}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{1}}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{10}\mathbf{,}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{2}}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{,}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{3}}\mathbf{\right)}\mathbf{=}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{4}}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{2}\mathbf{,}\\ \mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{5}}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{6}\mathbf{,}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{6}}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{3}\mathbf{,}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{7}}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{06}\mathbf{andP}\mathbf{\left(}{\mathbf{E}}_{\mathbf{8}}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{3}\end{array}$

Find each of the following probabilities:

$\begin{array}{l}\mathbf{a}\mathbf{.}\mathbf{P}\mathbf{\left(}{\mathbf{A}}^{\mathbf{c}}\mathbf{\right)}\\ \mathbf{b}\mathbf{.}\mathbf{P}\mathbf{\left(}{\mathbf{B}}^{\mathbf{c}}\mathbf{\right)}\\ \mathbf{c}\mathbf{.}\mathbf{P}\mathbf{(}{\mathbf{A}}^{\mathbf{c}}\mathbf{\cap }\mathbf{B}\mathbf{)}\\ \mathbf{d}\mathbf{.}\mathbf{P}\mathbf{(}\mathbf{A}\mathbf{\cup }\mathbf{B}\mathbf{)}\\ \mathbf{e}\mathbf{.}\mathbf{P}\mathbf{(}\mathbf{A}\mathbf{\cap }\mathbf{B}\mathbf{)}\\ \mathbf{f}\mathbf{.}\mathbf{P}\mathbf{(}{\mathbf{A}}^{\mathbf{c}}\mathbf{\cap }{\mathbf{B}}^{\mathbf{c}}\mathbf{)}\end{array}$

g. Are events A and B mutually exclusive? Why?

A Venn diagram is a probability diagram that shows logical relationships between events by placing one or more circles inside a rectangle. The rectangle in a Venn diagram represents the sample space, also known as the universal set, which collects all possible outcomes.

We know that probability $\left(\mathrm{x}\right)=\sum _{\mathrm{i}=1}^{\mathrm{x}}{\mathrm{x}}_{\mathrm{i}}$

Where,

role="math" localid="1653572732973" ${\mathbf{x}}_{\mathbf{i}}$are the events belonging to x.

So,

$\begin{array}{c}\mathbf{P}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{=}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{1}}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{2}}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{4}}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{5}}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{7}}\mathbf{\right)}\\ \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{10}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{2}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{06}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{06}\\ \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{47}\end{array}$

$$\begin{gathered} \mathbf{P}\mathbf{\left(}\mathbf{B}\mathbf{\right)}\mathbf{=}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{2}}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{3}}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{4}}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{5}}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{6}}\mathbf{\right)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{2}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{2}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{06}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{3} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{81} \end{gathered}$$

$$\begin{gathered} \mathbf{P}\mathbf{\left(}{\mathbf{A}}^{\mathbf{c}}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{-}\mathbf{P}\mathbf{\left(}\mathbf{A}\mathbf{\right)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{1}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{47} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{53} \end{gathered}$$

$$\begin{gathered} \mathbf{P}\mathbf{\left(}{\mathbf{B}}^{\mathbf{c}}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{-}\mathbf{P}\mathbf{\left(}\mathbf{B}\mathbf{\right)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{1}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{81} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{19} \end{gathered}$$

$\mathbf{P}\mathbf{(}{\mathbf{A}}^{\mathbf{c}}\mathbf{\cap }\mathbf{B}\mathbf{)}\mathbf{=}\mathbf{P}\mathbf{\left(}\mathbf{B}\mathbf{\right)}\mathbf{-}\mathbf{P}\mathbf{(}\mathbf{A}\mathbf{\cap }\mathbf{B}\mathbf{)}$

Here,

$\mathbf{P}\mathbf{\left(}\mathbf{B}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{81}$

$$\begin{gathered} \mathbf{P}\mathbf{(}\mathbf{A}\mathbf{\cap }\mathbf{B}\mathbf{)}\mathbf{=}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{2}}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{4}}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{5}}\mathbf{\right)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{2}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{6} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{85} \end{gathered}$$

Now, we get the value $\mathbf{P}\mathbf{(}{\mathbf{A}}^{\mathbf{c}}\mathbf{\cap }\mathbf{B}\mathbf{)}$

$$\begin{gathered} \mathbf{P}\mathbf{(}{\mathbf{A}}^{\mathbf{c}}\mathbf{\cap }\mathbf{B}\mathbf{)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{81}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{85} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{66} \end{gathered}$$

$$\begin{gathered} \mathbf{P}\mathbf{(}\mathbf{A}\mathbf{\cup }\mathbf{B}\mathbf{)}\mathbf{=}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{1}}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{2}}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{3}}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{4}}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{5}}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{6}}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{7}}\mathbf{\right)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{10}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{2}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{2}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{6}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{3}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{06} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{51} \end{gathered}$$

$$\begin{gathered} \mathbf{P}\mathbf{(}\mathbf{A}\mathbf{\cap }\mathbf{B}\mathbf{)}\mathbf{=}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{2}}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{4}}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}{\mathbf{E}}_{\mathbf{5}}\mathbf{\right)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{2}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{6} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{85} \end{gathered}$$

$$\begin{gathered} \mathbf{P}\mathbf{(}{\mathbf{A}}^{\mathbf{c}}\mathbf{\cap }{\mathbf{B}}^{\mathbf{c}}\mathbf{)}\mathbf{=}\mathbf{P}{\mathbf{(}\mathbf{A}\mathbf{\cup }\mathbf{B}\mathbf{)}}^{\mathbf{C}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{1}\mathbf{-}\mathbf{P}\mathbf{(}\mathbf{A}\mathbf{\cup }\mathbf{B}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{1}\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{51} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{51} \end{gathered}$$

A and B are not mutually exclusive occurrences.

Since,

$$\begin{gathered} \mathbf{P}\mathbf{(}\mathbf{A}\mathbf{\cap }\mathbf{B}\mathbf{)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{31}\mathbf{\ne }\mathbf{0} \\ \mathbf{P}\mathbf{(}\mathbf{A}\mathbf{\cap }\mathbf{B}\mathbf{)}\mathbf{\ne }\mathbf{0} \end{gathered}$$