Question

In 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?

A) Yes. P(S∩F) = P(S)·P(F) = 6%
B) No. P(S∩F) = 6% and P(S)·P(F) = 2.4%
C) No. P(S∩F) = 30% and P(S)·P(F) = 2.4%
D) No. P(S∩F) = 32% and P(S)·P(F) = 2.4%

Answer 1

The right answer is B

Answer 2

Option B - No. P(S∩F) = 6% and P(S)·P(F) = 2.4%

Explanation:

Given : In 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%.

To find : Are being a freshman and taking Spanish independent?

Solution :

Two events A and B are independent if

`P(A\cap B)=P(A)* P(B)`

We have given,

Total number of students = 1250

Students take Freshmen F = 250

Students take Spanish S= 150

`\text{Probability}=\frac{\text{Favorable outcome}}{\text{Total number of outcome}}`

`\text{P(F)}=(250)/(1250)`

`\text{P(S)}=(150)/(1250)`

`\text{P(S/F)}=30\%=(30)/(100)`

To show,
`P(S\cap F)=P(S)* P(F)`

Now, Taking LHS

`P(S\cap F)=P(F)* P(S/F)`

`P(S\cap F)=(250)/(1250)* (30)/(100)`

`P(S\cap F)=0.2* 0.3`

`P(S\cap F)=0.06`

`P(S\cap F)=6\%`

Now, Taking RHS

`P(S)* P(F)=(150)/(1250)* (250)/(1250)`

`P(S)* P(F)=0.12* 0.2`

`P(S)* P(F)=0.024`

`P(S)* P(F)=2.4\%`

Since,
`LHS\\eq RHS`

Being a freshman and taking Spanish are not independent.

Therefore, Option B is correct.

No. P(S∩F) = 6% and P(S)·P(F) = 2.4%