College Algebra, Chapter 10, 10.3, Section 10.3, Problem 46

A card is drawn at random from a standard 52-card deck. Determine whether the events $E$ and $F$ are mutually exclusive. Then find the
probability of the event $E \bigcup F$.

a.) $E:$ The card is a club.

$F:$ The card is a king.

In this case, the two events are not mutually exclusive since there is king of clubs. Thus, we have


$
\begin{equation}
\begin{aligned}

P(E \bigcup F) =& P(E) + P(F) - P(E \bigcap F)
\\
\\
=& \left( \frac{13}{52} \right) + \left( \frac{4}{52} \right) - \left( \frac{1}{52} \right)
\\
\\
=& \frac{16}{52}
\\
\\
=& \frac{4}{13}

\end{aligned}
\end{equation}
$



b.) $E:$ The card is an ace.

$F:$ The card is a spade.

Similarly, the two events are not mutually exclusive since there is an ace of spades. Thus, we have


$
\begin{equation}
\begin{aligned}

P(E \bigcup F) =& P(E) + P(F) - P(E \bigcap F)
\\
\\
=& \left( \frac{4}{52} \right) + \left( \frac{13}{52} \right) - \left( \frac{1}{52} \right)
\\
\\
=& \frac{16}{52}
\\
\\
=& \frac{4}{13}


\end{aligned}
\end{equation}
$