Given: A and B are two sets such that-
[tex]\begin{gathered} A\cap B=\phi \\ Pr(A)=0.3 \\ Pr(B)=0.4 \end{gathered}[/tex]
Required: To determine-
[tex]\begin{gathered} Pr(A\cap B) \\ Pr(A\cup B) \end{gathered}[/tex]
Explanation: Since A and B have no common elements, the events are independent events or disjoints or mutually exclusive.
For independent events, we have-
[tex]Pr(A\cap B)=Pr(A).Pr(B)[/tex]
Substituting the values into the formula-
[tex]\begin{gathered} Pr(A\cap B)=0.3\times0.4 \\ =0.12 \end{gathered}[/tex]
Recall that-
[tex]Pr(A\cup B)=Pr(A)+Pr(B)-Pr(A\cap B)[/tex]
Substituting the values into the formula and further solving as-
[tex]\begin{gathered} Pr(A\cup B)=0.3+0.4-0.12 \\ =0.7-0.12 \\ =0.58 \end{gathered}[/tex]
Final Answer: a)
[tex]Pr(A\cap B)=0.12[/tex]
b)
[tex]Pr(A\cup B)=0.58[/tex]
