a) In general, if two events, A and B, are independent,
[tex]\begin{gathered} A,B\rightarrow\text{ independent events} \\ \Rightarrow P(A\cap B)=P(A)*P(B) \end{gathered}[/tex]
Therefore, in our case,
[tex]\begin{gathered} P(Winning\cap Home)=0.2 \\ and \\ P(Winning)*P(Home)=0.25*0.8=0.2 \end{gathered}[/tex]
Thus, the two events are independent.
b) Similarly,
[tex]\begin{gathered} P(losing\cap away)=0.15 \\ and \\ P(losing)P(away)=0.75*0.20=0.15 \end{gathered}[/tex]
The two events are independent.
Furthermore, the intersection of two independent events is different than the empty set.
