Question

# List all events associated with the random experiment of tossing of two coins. How many of them are elementary events ?

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Hint: First, find out the sample space that contains all the possible outcomes of this random
experiment. Then proceed on to finding which of them are elementary events. Elementary events are
events that aren’t subsets derived from any other events related to the random experiment.

Let’s first analyse the experiment given to us. The experiment is the tossing of two coins, simultaneously.
Now, since the two coins are tossed simultaneously, there can be outcomes of heads or tails for both
the coins.
However, the outcomes of both the coins together will be considered as a possibility.
This tells us, that the possibilities are as follows :
1. The first coin shows heads, and the second coin shows heads too.
2. The first coin shows heads, and the second coin shows tails.
3. The first coin shows tails, and the second coin shows heads.
4. The first coin shows tails, and the second coin shows tails as well.
Therefore, these are all the possible outcomes. All these possible outcomes will form the sample space
for this event.
Thus, the sample space S for this event can be written as follows :
$S=\{(H,H),(H,T),(T,H),(T,T)\}$
Thus, these four outcomes are all the possibilities that can come out of tossing two coins.
Now, the next part of the question asks us how many of these events are elementary. For this, we first
need to understand what exactly is an elementary event.
An elementary event is an event, which is not derived from any other event. For example, here, our
possible events are the sample space $S$, as shown. However, we can derive more subsets from the
sample space $S$, which contains the sample space, using set theory.
Therefore, all the events in the sample space $S$ are called elementary events. Hence, all four of the
possible outcomes are elementary events in this case.
The possible outcomes are $S=\{(H,H),(H,T),(T,H),(T,T)\}$, and all the events in $S$ are elementary
events.

Note: To understand the concept of elementary events better, consider the tossing of a single coin. The
sample space for the tossing of a single coin is $S=\{(H),(T)\}$ since we are only tossing one coin.
However, from here can derive four subsets, using set theory. Therefore, the subsets of $S$ are : {H}, {T},
{H,T}, and $\phi$. The number of subsets that are derived from a sample space $S$ = ${{2}^{n}}$, where
$n$ is the number of elements in the sample space $S$.
Hence, elementary events are the events that are the possible outcomes of a random experiment, and
are not the subsets of any other outcomes of the experiment.