Assume 75% of the AP stats students studied for this test. If 40% of those who studied get an A but only 10% of those who did not study get A, what is the probability that someone who gets an A actually studied for the test?
2 Answers
Let
Let
We know that
We know also that
We are looking for
The probability is (approximately) 92.31%.
Explanation:
Let
Let
It sounds like we are seeking the conditional probability of, "studied for the test GIVEN THAT they got an A".
This is written as
We are told:
#"P"(B) = 0.75# which means#"P"(B^"C") = 0.25#
#"P"(A|B)=0.4#
#"P"(A|B^"C") = 0.1#
(the#""^"C"# means "complement")
and we want to find
#"P"(B|A) = ("P"(B nn A))/("P"(A))#
Using Bayes' Theorem, we can rewrite this as
#"P"(B|A) = ("P"(A|B)"P"(B))/("P"(A|B) "P"(B)+"P"(A|B^"C") "P"(B^"C"))#
Every probability on the right side is now something we know. We plug these values in:
#"P"(B|A) = (0.4 xx 0.75)/(0.4 xx 0.75" " +" "0.1 xx 0.25)#
#color(white)("P"(B|A)) = (0.3)/(0.3+0.025)#
#color(white)("P"(B|A)) = (0.3)/(0.325)#
#color(white)("P"(B|A)) = 12/13" " ~~ 0.9231#
So there is a 92.31% chance that, given that a student got an A, they also studied.