2 Suppose among the students registered in a course 30 are s

2. Suppose among the students registered in a course, 30% are social science majors, 20% are business majors, and the remaining 50% are other majors. Among each of these groups, 40% are first year students, 30% are second year students and the remaining 30% are upper year students.

(a) Define which events are mutually exclusive.

(b) What is the probability that a randomly chosen student is a second year business major?

(c) What is the probability that a randomly chose student is either a second year, or a business major (he/she could be both)?

Solution

(a)

Events that student registered in a course social science majors, or business majors, or other majors are mutually exclusive events.

(b)

Let S shows the event that students registered in a course social science major, B shows the event that students registered in a course business majors and O shows the event that students registered in a course other majors. Let F shows the event that student is in first year, E shows the event that student is in second year and U shows the event that student is an upper year student. So we have following probabilties:

P(S)=0.30, P(B)=0.20, P(O)=0.50, P(F|S)=0.40, P(E|S)=0.30, P(U|S)=0.30

P(F|B)=0.40, P(E|B)=0.30, P(U|B)=0.30,P(F|O)=0.40, P(E|O)=0.30, P(U|O)=0.30

So the required probability is

P(B and E)=P(E|B)P(B)=0.30*0.20=0.06

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(c)

By the total law of probability we have:

P(E)=P(E|B)P(B)+P(E|S)P(S)+P(E|O)P(O)=0.30*0.20+0.30*0.30+0.30*0.50=0.06+0.09+0.15=0.3

So the probability that a randomly chose student is either a second year, or a business major (he/she could be both) is

P(E or B)=P(E)+P(B)-P(E and B)=0.3+0.2-0.06=0.44