Question

In Exercises 5–36, express all probabilities as fractions.

Sorting Hat At Hogwarts School of Witchcraft and Wizardry, the Sorting Hat chooses one of four houses for each first-year student. If 4 students are randomly selected from 16 available students (including Harry Potter), what is the probability that they are the four youngest students?

Short answer

The probability of selecting the four youngest students is equal to $\frac{1}{1820}$.

Step-by-step solution

Given information

There are 16 students. Four are to be selected for allotting houses.

Define combination

The formula of combination is used to estimate the number of ways in which r items can be selected from n items without replacement when the order of the items does not matter.

${}^n{C}_r=\frac{n!}{\left(n-r\right)!r!}$

Calculation

Let A be the event of selecting the youngest four students.

The total number of ways in which four students can be selected from 16 students (in any order) is computed below:

$$\begin{gathered} {}^{16}{C}_4=\frac{16!}{\left(16-4\right)!4!} \\ =1820 \end{gathered}$$

The number of favorable ways in which the four youngest students can be selected is one.

The probability of selecting the four youngest students is shown below:

$$\begin{gathered} P\left(A\right)=\frac{\text{Number}\text{of}\text{favourable}\text{ways}}{\text{Total}\text{number}\text{of}\text{ways}} \\ =\frac{1}{1820} \end{gathered}$$

Therefore, the probability of selecting the four youngest students is $\frac{1}{1820}$.