4155 The English majors at a certain university revealed the

4.155 The English majors at a certain university revealed the following information: 10 percent of them failed mathematics, 20 percent failed biology, and 5 percent failed both mathematics and biology. Determine the probability of the following events:

An English major failed mathematics, given that he or she passed biology.

The student passed biology, given that he or she passed mathematics.

The student passed mathematics and it is known that he or she failed biology.

d)         The student passes both courses

Solution

let A denote an event that a student fails in Mathematics i.e P(A) = 0.1

let B denote an event that a student fails in Biology i.e P(B) = 0.2

P(AnB) 0.05

Let A^c denote an event that a student passes in Mathematics i.e P(A^c) = 0.9

Let B^c denote an event that a student passes in biology i.e P(B^c) = 0.8

a) Probability that a English major failed mathematics, given that he or she passed biology.

= P(A|B^c) = P(AnB^c) / P(B^c)

P(AnB^c) = P(A) - P(AnB) = 0.1 - 0.05 = 0.05

therefore, P(A|B^c) = P(AnB^c) / P(B^c) = 0.05/ 0.8 = 0.0625

Probability that a English major failed mathematics, given that he or she passed biology is 0.0625

b) Probability that the student passed biology, given that he or she passed mathematics is

P(B^c|A^c ) = P(B^cnA^c ) | P(A^c )

P(B^cnA^c ) = P(AuB)^c = 1 - P(AuB) = 1 - [P(A) + P(B) - P(AnB)] = 1 - [0.1 +0.2 -0.05] = 1 - 0.25 = 0.75

P(B^cnA^c ) | P(A^c ) = 0.75/ 0.9 = 0.8333

c) Probability that student passed mathematics and it is known that he or she failed biology

P(A^c|B) = P(A^cnB) | P(B)

P(A^cnB) = P(B) - P(AnB) = 0.2 -0.05 = 0.15

P(A^cnB) | P(B) = 0.15/0.2 = 0.75

d) Probability that students passes both subject

P(A^cn B^c) = 0.75 as computed above