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

Answer:

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

Step-by-step 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:

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

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:

$P(Y|X)=\frac{P(X|Y)P(Y)}{P(X|Y)P(Y)+P(X|Z)P(Z)}$

             $=\frac{(0.32\times 0.36)}{(0.32\times 0.36)+(0.68\times 0.18)}$

             $=0.4848$

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