There are two identical dice with a single letter on each of the faces. The following six letters : Q, R, S,T, U and 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 : Q, U and V?

The correct answer is 1/4.

To find the probability that the outcomes on both dice are composed only of any combination of the letters Q, U, and V, we can analyze the problem step-by-step:

Step 1: Identify the total number of outcomes for a single die.
Each die has six faces, each containing one of the following six letters: Q, R, S, T, U, and V.
Therefore, the total number of possible outcomes when rolling a single die is:

$\text{Total outcomes}=6$

Step 2: Identify the favorable outcomes for a single die.
We want the outcomes to consist only of the letters Q, U, or V. These are our favorable outcomes.
The number of favorable letters is 3 (namely Q, U, and V).
Therefore, the number of favorable outcomes for a single die is:

$\text{Favorable outcomes}=3$

Step 3: Calculate the probability for each die.
Since the outcomes are equally likely, the probability of rolling Q, U, or V on the first die is:

$P\left({\text{Die}}_1\right)=\frac{3}{6}=\frac{1}{2}$

$P\left({\text{Die}}_2\right)=\frac{3}{6}=\frac{1}{2}$

Step 4: Calculate the joint probability.
Since the two dice are thrown independently at random, we multiply the individual probabilities to find the combined probability of both events occurring:

$\text{Total Probability}=P\left({\text{Die}}_1\right)\times P\left({\text{Die}}_2\right)$

$\text{Total Probability}=\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}$