Question

Three coins are tossed simultaneously, find S and n(S).

Answer 1

Hint : When a coin is tossed, then there are only two possibilities of events that are possible, which means the result of this event is either “Heads” or “Tails.” In this question, we have to find the sample space of a coin tossed. The above problem can be resolved by taking the sample space of two coins. Here S is denoted for sample space and n(S) denotes for the number of sample space formed in a sample.
Formula used:
To find the number of sample space, we use the below formula,
$n\left( S \right) = {2^x}$
Here, $x$ is the number of coins.

Complete step-by-step answer :
We have to first write the outcomes of a single coin when it is tossed,
$S = \\{ \left( H \right)\left( T \right)\\}$
Here, H is denoted the head and T is denoted the tails.
There are only two possibilities, either head or tail.
Now, tossed two coins, we get the sample space,
S = \left\\{ {\left( {H,T} \right)\left( {T,H} \right)} \right\\}
There are two possibilities.
Now, if we tossed three coins simultaneously, we get the sample space,
S = \left\\{ {\left( {HHT} \right),\left( {HHT} \right),\left( {HTH} \right),\left( {HTT} \right),\left( {THH} \right),\left( {THT} \right),\left( {TTH} \right),\left( {TTT} \right)} \right\\}
There are a total of eight possibilities.

Now we use the formula to find the number of sample spaces.
The expression for the number of sample space we use the below formula,
$n\left( S \right) = {2^x}$
Three coins are used, so substitute $3$ for $x$ in the above expression we get,
$\begin{array}{l} n\left( S \right) = {2^3}\\\ n\left( S \right) = 8 \end{array}$
Thus, the number of sample spaces is $8$ .

Note : To solve these types of questions, we must have knowledge of outcomes. If a coin is tossed it will give either only “Heads” or “Tails.” If three coins are tossed simultaneously, then make three possible outcomes. We can also first find the number of sample spaces formed by using this formula $n\left( S \right) = {2^x}$ . So we can get the idea regarding the number of sample spaces formed. So, we can cross-check our outcomes.