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?

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

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virtuematane virtuematane · Answer 2 · 2019-05-21T09:25:58+02:00

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:

[tex]P(A\bigcap B)=P(A)\times P(B)[/tex]

Also, in such a condition we have:

[tex]P(B|A)=P(B)[/tex]

Now here we have:

[tex]P(A)=\dfrac{250}{1250}=\dfrac{1}{5}\\\\\\P(B)=\dfrac{150}{1250}=\dfrac{3}{25}[/tex]

Also we are given,

[tex]P(B|A)=0.30=\dfrac{3}{10}[/tex]

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

As we see that:

[tex]P(B|A)\neq P(B)[/tex]

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