Question

How do you find the probability of combinations?

Answer 1

Probability of Combinations

We use combinations to calculate the total number of outcomes of an event where the order of the outcomes does not matter. The formula to calculate the combinations is given by:

${}^{\mathbf{n}}\mathit{C}_{\mathbf{r}}\mathbf{}\mathbf{=}\frac{\mathbf{n}\mathbf{!}}{\mathbf{(}\mathbf{n}\mathbf{-}\mathbf{r}\mathbf{)}\mathbf{!}\mathbf{r}\mathbf{!}}$

Here,

$n$ = Total number of items

$r$ = Number of items to be chosen at a time

In probability problems, combinations are used to find the number of favorable and total outcomes of an event. Consider the following example.

In a team, $4$ girls and $3$ boys are there. Among them, $4$ members need to be selected for one round of a game. Find the probability of selecting an equal number of boys and girls chosen for the round of the game.

Probability of selecting an equal number of boys and girls

$$\begin{gathered} =\frac{4C_2\times 3C_2}{7C_4} \\ =\frac{18}{35} \end{gathered}$$

Hence, probability of selecting an equal number of boys and girls is $\frac{\mathbf{18}}{\mathbf{35}}$.