Answer
$\dfrac{1}{12}$
Work Step by Step
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,
$P(E)=P(E_1)+P(E_2)=\dfrac{1}{24}+\dfrac{1}{24}=\dfrac{2}{24}=\dfrac{1}{12}$
