Question

From a group of 2 boys and 3 girls, two children are selected. Find the sample space of this experiment.

Answer 1

Hint: First, we calculate the total number of possible ways to select 2 children at random using a method of combination. We form sample space for the given experiment by collecting each possibility of pairing two children be it both girls, both boys or a girl and a boy. Denote two boys as subscripts of B and three girls as subscripts of G. Make possible pairs of children.
* Combination is given by \[^n{C_r} = \dfrac{{n!}}{{(n - r)!r!}}\]where factorial opens up as \[n! = n(n - 1)! = n(n - 1)(n - 2)!....\]
* Sample space of an experiment is the set of all possible outcomes of a random experiment.

Complete step-by-step answer:
We are given there are 2 boys and 3 girls.
So, the total number of children is 5.
We first calculate the total number of ways to select 2 children from the total 5 children using the method of combination.
Number of ways of selecting 2 children out of 5 is \[^5{C_2}\].
Here \[n = 5,r = 2\]
\[{ \Rightarrow ^5}{C_2} = \dfrac{{5!}}{{(5 - 2)!2!}}\]
\[{ \Rightarrow ^5}{C_2} = \dfrac{{5 \times 4 \times 3!}}{{3!2!}}\]
Cancel same factors from numerator and denominator,
\[{ \Rightarrow ^5}{C_2} = \dfrac{{5 \times 4}}{2}\]
Cancel same factors from numerator and denominator,
\[{ \Rightarrow ^5}{C_2} = 10\]
So, there are 10 possible ways of selecting 2 children at random from a total 5 children.
Now let us assume 2 boys as \[{B_1},{B_2}\] and 3 girls as \[{G_1},{G_2},{G_3}\]
We form 3 cases: both boys, both girls and one boy one girl,
CASE 1: Both boys
There are 2 boys \[{B_1},{B_2}\]
We can only take one possibility when we choose both boys from 5 children i.e. \[{B_1}{B_2}\] … (1)
CASE 2: Both girls
There are 3 girls \[{G_1},{G_2},{G_3}\]
Possible combinations are: \[{G_1}{G_2}\]; \[{G_2}{G_3}\]and \[{G_1}{G_3}\] … (2)
CASE 3: One boy one girl
There are 3 girls \[{G_1},{G_2},{G_3}\]and 2 boys \[{B_1},{B_2}\]
Possible combinations are: \[{B_1}{G_1}\]; \[{B_1}{G_2}\]; \[{B_1}{G_3}\]; \[{B_2}{G_1}\]; \[{B_2}{G_2}\]and \[{B_2}{G_3}\] … (3)
Now we combine all the possibilities to write the sample space for the experiment.
Let sample space of the experiment be denoted by S.
\[ \Rightarrow S = \left\{ {{B_1}{B_2},{G_1}{G_2},{G_2}{G_3},{G_1}{G_3},{B_1}{G_1},{B_1}{G_2},{B_1}{G_3},{B_2}{G_1},{B_2}{G_2},{B_2}{G_3}} \right\}\]

\[\therefore \]Sample space for experiment is \[\left\{ {{B_1}{B_2},{G_1}{G_2},{G_2}{G_3},{G_1}{G_3},{B_1}{G_1},{B_1}{G_2},{B_1}{G_3},{B_2}{G_1},{B_2}{G_2},{B_2}{G_3}} \right\}\]

Note:
Many students make the mistake of writing the sample space with only three possibilities, each of a case which is wrong, even though we are not given which child we have to choose we have choices between which girl and which boy is chosen.