Question

There are three coloured dice of red, white and black. These dice are placed in a bag. One die is drawn at random from the bag and rolled, its colour and the uppermost face is noted. Describe the sample space for this experiment.

Answer 1

Hint: We solve this question by first going through the definition of the Sample Space. Then we find the possible outcomes of drawing a die from the bag. Then we find the possible outcomes of rolling a die. Then we combine the both results for every case and find the Sample space of the given experiment.

Complete step by step answer:
We are given that there are three coloured dice of Red, White and Black. They are placed in a bag and one die is picked at random.
The die that is picked is rolled and the colour of the die and the number on the uppermost face of the die are noted.
We need to find the Sample Space of the experiment.
First let us go through the definition of a Sample Space.
The set of all possible outcomes for an experiment is called the Sample Space of the experiment.
So, to find the sample space of our experiment, we need to find all possible outcomes of the experiment.
In the experiment, first we are drawing a die from the bag. So, the possible outcomes are
$\left\{ \text{Red,White,Black} \right\}$
Then the die that is drawn from the bag is rolled. For any die it has six faces with numbers 1 to 6 on it. So, the possible outcomes when a die is rolled are
$\left\{ 1,2,3,4,5,6 \right\}$
We are given that after rolling the dice the colour of the dice and the number on the upper face of the die is noted. So, we need to combine the both sets above.
So, we get the possible outcomes as
$\left\{ \begin{align}
  & \left( \text{Red,1} \right),\left( \text{Red,2} \right),\left( \text{Red,3} \right),\left( \text{Red,4} \right),\left( \text{Red,5} \right),\left( \text{Red,6} \right), \\
 & \left( \text{White,1} \right),\left( \text{White,2} \right),\left( \text{White,3} \right),\left( \text{White,4} \right),\left( \text{White,5} \right),\left( \text{White,6} \right) \\
 & \left( \text{Black,1} \right),\left( \text{Black,2} \right),\left( \text{Black,3} \right),\left( \text{Black,4} \right),\left( \text{Black,5} \right),\left( \text{Black,6} \right) \\
\end{align} \right\}$
So, we get the sample space of the experiment as
$\left\{ \begin{align}
  & \left( \text{Red,1} \right),\left( \text{Red,2} \right),\left( \text{Red,3} \right),\left( \text{Red,4} \right),\left( \text{Red,5} \right),\left( \text{Red,6} \right), \\
 & \left( \text{White,1} \right),\left( \text{White,2} \right),\left( \text{White,3} \right),\left( \text{White,4} \right),\left( \text{White,5} \right),\left( \text{White,6} \right) \\
 & \left( \text{Black,1} \right),\left( \text{Black,2} \right),\left( \text{Black,3} \right),\left( \text{Black,4} \right),\left( \text{Black,5} \right),\left( \text{Black,6} \right) \\
\end{align} \right\}$

Hence answer is $\left\{ \begin{align}
  & \left( \text{Red,1} \right),\left( \text{Red,2} \right),\left( \text{Red,3} \right),\left( \text{Red,4} \right),\left( \text{Red,5} \right),\left( \text{Red,6} \right), \\
 & \left( \text{White,1} \right),\left( \text{White,2} \right),\left( \text{White,3} \right),\left( \text{White,4} \right),\left( \text{White,5} \right),\left( \text{White,6} \right) \\
 & \left( \text{Black,1} \right),\left( \text{Black,2} \right),\left( \text{Black,3} \right),\left( \text{Black,4} \right),\left( \text{Black,5} \right),\left( \text{Black,6} \right) \\
\end{align} \right\}$.

Note:
The common mistake one makes is while combining the sets $\left\{ \text{Red,White,Black} \right\}$ and $\left\{ 1,2,3,4,5,6 \right\}$, one might combine them and write the possible outcomes as $\left\{ \text{Red,White,Black},1,2,3,4,5,6 \right\}$. But here we need consider the cases while die is red then the possible outcomes of die and then when the die is white then the possible outcomes of die and similarly when the die is black.