Question

n a school of 1250 students, 250 are freshmen and 150 students take Spanish. The probability that a student takes Spanish given that he/she is a freshman is 30%. Are being a freshman and taking Spanish independent?

Answer 1

Hello :)

Beimg a freshman and taking Spanish are independent. This is the case because you do not need to be a freshman to take Spanish.

Hope this helps and have a great day :)

Answer 2

Answer:

No, being a freshman and taking Spanish are not independent.

Step-by-step explanation:

Let A denote the event that the student is a freshman.

B denote the event that the student is taking Spanish.

Let A∩B denote the event that the the student is a freshman and takes Spanish.

Let P denote the probability of an event.

Now we know that:

Two events A and B are said to be independent if:

        $P(A\bigcap B)=P(A)\times P(B)$

Also, in such a condition we have:

$P(B|A)=P(B)$

Now here we have:

$P(A)=\dfrac{250}{1250}=\dfrac{1}{5}\\\\\\P(B)=\dfrac{150}{1250}=\dfrac{3}{25}$

Also we are given,

$P(B|A)=0.30=\dfrac{3}{10}$

( Since, we are given  The probability that a student takes Spanish given that he/she is a freshman is 30% )

As we see that:

$P(B|A)\neq P(B)$

Hence, being a freshman and taking Spanish are not independent.