There are two identical dice with a single letter on each of the faces. The following six letters: $\text{Q, R, S, T, U}$ and $\text{V}$, one on each of the faces. Any of the six outcomes are equally likely.

The two dice are thrown once independently at random.

What is the probability that the outcomes on the dice were composed only of any combination of the following possible outcomes: $\text{Q, U}$ and $\text{V}$?

1. $\frac{1}{4}$
2. $\frac{3}{4}$
3. $\frac{1}{6}$
4. $\frac{5}{36}$

Let the sample space of dice $1$ and dice $2$ be $\text{S}_{1}$ and $\text{S}_{2}$ respectively.

• ${\color{Blue}{\text{S}_{1} = \{\text{Q, R, S, T, U, V}\} = 6}}$
• ${\color{Purple}{\text{S}_{2} = \{\text{Q, R, S, T, U, V}\} = 6}}$

The two dice are thrown once independently at random.

In probability, two events are ${\color{Magenta}{\text{independent}}}$ if the incidence of one event does not affect the probability of the other event. If the incidence of one event does affect the probability of the other event, then the events are ${\color{Red}{\text{dependent}}}$.

• Two events $\text{A}$ and $\text{B}$ are ${\color{Magenta}{\text{independent}}}$ if:
• $\text{P} (\text{A} \cap \text{B}) = \text{P(A)} \cdot \text{P(B)}$

The probability that the outcomes on the dice were composed only of any combination of the following possible outcomes$: \text{Q, U},$ and $\text{V} = \frac{3}{6} \times \frac{3}{6} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}.$

Correct Answer $:\text{A}$

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