Question

In fall 2014, 36% of applicants with a Math SAT of 700 or more were admitted by a certain university, while 18% with a Math SAT of less than 700 were admitted. Further, 32% of all applicants had a Math SAT score of 700 or more. What percentage of admitted applicants had a Math SAT of 700 or more

Answer 1

The percentage of admitted applicants who had a Math SAT of 700 or more is 48.48%.

Explanation:

The Bayes' theorem is used to determine the conditional probability of an event E
`_(i)`, belonging to the sample space S = (E₁, E₂, E₃,...Eₙ) given that another event A has already occurred by the formula:

`P(E_(i)|A)=\fracE_(i))P(E_(i)){\sum\limits^(n)_(i=1)E_(i))P(E_(i))}`

Denote the events as follows:

X = an student with a Math SAT of 700 or more applied for the college

Y = an applicant with a Math SAT of 700 or more was admitted

Z = an applicant with a Math SAT of less than 700 was admitted

The information provided is:

`P(Y)=0.36\\P(Z)=0.18\\P(X|Y)=0.32`

Compute the value of
`P(X|Z)` as follows:

`P(X|Z)=1-P(X|Y)\\=1-0.32\\=0.68`

Compute the value of P (Y|X) as follows:

`P(Y|X)=(P(X|Y)P(Y))/(P(X|Y)P(Y)+P(X|Z)P(Z))`

`=((0.32* 0.36))/((0.32* 0.36)+(0.68* 0.18))`

`=0.4848`

Thus, the percentage of admitted applicants who had a Math SAT of 700 or more is 48.48%.