Question

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

Answer 1

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.