College Algebra, Chapter 10, 10.3, Section 10.3, Problem 54

A coin is tossed twice. Let $E$ and $F$ be the following events:

$E:$ The first toss shows heads.

$F:$ The second toss shows heads.

a.) Are the events $E$ and $F$ independent?

Yes, because the occurrence of one event doesn't affect the probability of another event.

b.) Find the probability of showing heads on both tosses.

If the events $E$ and $F$ are independent in a sample, then the probability of $E$ and $F$ is


$\begin{equation} \begin{aligned} P(E \bigcap F) =& P(E) P(F) \\ \\ =& \left( \frac{1}{2} \right) \left( \frac{1}{2} \right) \\ \\ =& \frac{1}{4} \\ \\ =& 0.25 \end{aligned} \end{equation}$