Question

A sample of $300$students is taken. Of the students surveyed,$50$were music students, while $250$were not. Ninetyseven were on the honor roll, while$203$ were not. If we assume being a music student and being on the honor roll are independent events, what is the expected number of music students who are also on the honor roll?

Short answer

Around $16$kids from a sample are likely to be music students who are also on the honour roll.

Step-by-step solution

Given information

Given in the question that, A sample of $300$students is taken. Of the students surveyed, $50$were music students, while $250$were not. Ninetyseven were on the honor roll, while $203$were not. If we assume being a music student and being on the honor roll are independent events. We need to find the expected number of music students who are also on the honor roll if we assume being a music student and being on the honor roll are independent events.

Explanation

A total of $300$pupils are chosen as a sample. $50$of the students polled were music students, whereas the other $250$were not. Ninety-seven students were on the honour roll, while the remaining $203$were not.

Calculation:

Let the random variable $X$be defined as

$X=$Expected number of students on the honour roll who are also music students.

Let$A$represent the fact that the student is a music student, and $B$represent the fact that he or she is also on honour roll. We have the following condition if both occurrences $A$and$B$ are independent events:

$P(AANDB)=P\left(A\right)·P\left(B\right)$

We have,

$P\left(A\right)=\frac{50}{300}$

$P\left(B\right)=\frac{97}{300}$

$\Rightarrow \frac{X}{300}=\frac{50}{300}\times \frac{97}{300}$

localid="1653556696458" $$\begin{gathered} \Rightarrow X=\frac{4850}{300} \\ =16.17 \\ \approx 16 \end{gathered}$$

$\therefore X=16$