#### Answer

The probability is $\dfrac{1}{12}$.
Refer to the step-by-step part bellow for the model.

#### Work Step by Step

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