ABSTRACT
To introduce the notion of conditional probability, let us first examine the following question: Suppose that all of the freshmen of an engineering college took calculus and discrete math last semester. Suppose that 70% of the students passed calculus, 55% passed discrete math, and 45% passed both. If a randomly selected freshman is found to have passed calculus last semester, what is the probability that he or she also passed discrete math last semester? To answer this question, let A and B be the events that the randomly selected freshman passed discrete math and calculus last semester, respectively. Note that the quantity we are asked to find is not P(A), which is 0.55; it would have been if we were not aware that B has occurred. Knowing that B has occurred changes the chances of the occurrence of A. To find the desired probability, denoted by the symbol P ( A ∣ B ) $ P(A \mid B) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/inline-math3_1.tif"/> [read: probability of A given B] and called the conditional probability of A given B, let n be the number of all the freshmen in the engineering college. Then (0.7)n is the number of freshmen who passed calculus, and (0.45)n is the number of those who passed both calculus and discrete math. Therefore, of the (0.7)n freshmen who passed calculus, (0.45)n of them passed discrete math as well. It is given that the randomly selected student is one of the (0.7)n who passed calculus; we also want to find the probability that he or she is one of the (0.45)n who passed discrete math. This is obviously equal to ( 0.45 ) n / ( 0.7 ) n = 0.45 / 0.7 $ (0.45)n/(0.7)n =0.45/0.7 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/inline-math3_2.tif"/> . Hence P ( A ∣ B ) = 0.45 0.7 . $$ \begin{aligned} P(A \mid B) = \frac{0.45}{0.7}. \end{aligned} $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/um158.tif"/>
