Question

An experiment consists of rolling a die and then tossing a coin once if the number on the die is even. If the number on the die is odd the coin is tossed twice. Write the sample space for this experiment.

Answer 1

Hint: Try to figure out the sample space of a dice and coin tossed once and twice. Now combine the elements of even numbers in dice with a coin tossed once. And similarly, combine the odd numbers on dice with the coin tossed twice. Remember that the number of elements in sample space should be the product of the number of elements from tossing the coin and rolling dice.

Complete step-by-step answer:
Let’s try to analyse the question properly first. In an experiment, a dice is rolled and if the number on top is even, a coin is tossed once. And if the number on the top of dice is odd, then the coin is tossed twice.
With this information, we need to find the sample space for this experiment. But first, you need to understand what exactly a sample space for an experiment is. When dealing with any type of probability question, the sample space represents the set or collection of all possible outcomes. In other words, it is a list of every possible result when running the experiment just once.
This implies sample space for the rolling of dice will be a set of all the possible numbers on the dice, i.e. $\left\{ {1,2,3,4,5,6} \right\}$
And the sample space for the toss of a coin once will be head (H) or tail (T). So, the sample space for tossing a coin twice will be in pairs, i.e. HH, TT, HT, and TH.
For even numbers on the dice, the sample space will be made with one dice and one coin. But for odd numbers on the dice the sample space will be made with one dice and two coins.
$ \Rightarrow $For even numbers on the dice: $\left\{ {2H,2T,4H,4T,6T,6H} \right\}$
$ \Rightarrow $For odd numbers on the dice: $\left\{ {1HH,1HT,1TT,1TH,3HH,3HT,3TT,3TH,5HH,5HT,5TT,5TH,} \right\}$
Therefore, the sample space of the experiment will be: $\left\{ {1HH,1HT,1TT,1TH,3HH,3HT,3TT,3TH,5HH,5HT,5TT,5TH,2H,2T,4H,4T,6T,6H} \right\}$

Note: Be careful while writing the sample space with odd numbers. Notice that the position of H and T matters when the coin is tossed twice, just like it will matter in reality. Don’t forget to consider that while making the sample space. Also, note that sample space is a set, so should always be represented as a set only, i.e. using braces $'\{ \} '$