## Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

$\dfrac{1}{12}$
The experiment involves spinning $\text{Spinner II}$ followed by $\text{Spinner I}$ then finally by $\text{ Spinner III}$. Thus, the sample space $S$ is: $\left\{\text{Yellow 1 Forward }, \text{Yellow 1 Backward}, \text{Yellow 2 Forward }, \text{Yellow 2 Backward}, \text{Yellow 3 Forward}, \text{Yellow 3 Backward}, \text{Yellow 4 Forward}, \text{Yellow 4 Backward}, \text{Green1 Forward }, \text{Green 1 Backward}, \text{Green 2 Forward }, \text{Green 2 Backward}, \text{Green 3 Forward}, \text{Green 3 Backward}, \text{Green 4 Forward}, \text{Green 4 Backward}, \text{Red 1 Forward }, \text{Red 1 Backward}, \text{Red 2 Forward }, \text{Red 2 Backward}, \text{Red 3 Forward}, \text{Red 3 Backward}, \text{Red 4 Forward}, \text{Red 4 Backward} \right\}$ Note the sample space has $24$ equally-likely outcomes so $n(S)=24$. Let $E_1$ be the event that the outcome is a Yellow followed by a $2$. then by Forward. Then, from the sample space, we have $P(E_1)=\dfrac{1}{24}$. Let $E_2$ be the event that the outcome is a Yellow followed by a $4$, then by a Forward. Then, $P(E_2)=\dfrac{1}{24}$. Let $E$ = event that a Yellow comes out followed by a $2$ or a $4$, then by a Forward. Then, $P(E)=P(E_1)+P(E_2)=\dfrac{1}{24}+\dfrac{1}{24}=\dfrac{2}{24}=\dfrac{1}{12}$