Question

A coin is tossed. Write the sample space.

Answer 1

In any experiment, the sample space is a set of all the possible outcomes. While we already know that a coin has only two faces i.e. head and tail, the number of elements in the sample space depends on the number of times the coin is tossed.

Complete step by step solution:
According to the question, we have to write the sample space of an experiment where a coin is tossed.
We know as per the definition, the sample space of an experiment is a set of all the possible outcomes. Let this set be denoted by $S$.
We also know that a normal coin has only two faces i.e. head and tail (represented as $H$ and $T$). The number of elements in the sample space depends on the number of times the coin is tossed.
If the coin is tossed only once then the sample space ($S$) has only two possible outcomes (head and tail). So the sample space set for this is shown below:
$\Rightarrow S = \\{ H,T\\}$
If the coin is tossed twice we can have multiple outcomes as given below:
(1) First toss results in the head and the second toss also results in the head.
(2) First toss results in the head and the second toss results in the tail.
(3) First toss results in a tail and the second toss results in the head.
(4) And the first toss results in a tail and the second toss also results in a tail.
Writing these outcomes in set form, our sample space will be as shown:
$\Rightarrow S = \\{ \left( {H,H} \right),\left( {H,T} \right),\left( {T,H} \right),\left( {T,T} \right)\\}$
Similarly if the coin is tossed thrice then the sample space will be:
$\Rightarrow S = \\{ \left( {H,H,H} \right),\left( {H,H,T} \right),\left( {H,T,H} \right),\left( {H,T,T} \right),\left( {T,H,H} \right),\left( {T,H,T} \right),\left( {T,T,H} \right),\left( {T,T,T} \right)\\}$
In this way we can determine the sample space if the coin is tossed any number of times.

Note: If a normal coin is tossed any number of times then we have a formula to determine the total number of possible outcomes or the number of elements in the sample space.
Let $N$ is the number of elements in the sample space which also represents the total number of possible outcomes and $n$ is the number of times the coin is tossed. Then the formula is:
$\Rightarrow N = {2^n}$