Porisma

Worked solution

Distinguish dependence from a nonempty overlap

Two events A and B satisfy P(A)=1/4P(A)=1/4, P(A∣B)=1/2P(A\mid B)=1/2 and P(B∣A)=2/3P(B\mid A)=2/3.

(i) Determine whether A and B are independent, justifying your answer.

(ii) Determine whether A and B are mutually exclusive, justifying your answer.

(iii) Find P(B).

CJC · Lecture · Example 3

(i) Compare a conditional with its original probability

P(A∣B)=1/2≠P(A)=1/4P(A\mid B)=1/2\ne P(A)=1/4, so A and B are not independent.

How did this part go?

(ii) Calculate the intersection

From P(B∣A)=P(B∩A)/P(A)P(B\mid A)=P(B\cap A)/P(A), (2/3)=P(B∩A)/(1/4)(2/3)=P(B\cap A)/(1/4). Thus P(B∩A)=(2/3)(1/4)=1/6P(B\cap A)=(2/3)(1/4)=1/6. It is nonzero, so the events are not mutually exclusive.

How did this part go?

(iii) Use the other conditional

P(A∣B)=P(A∩B)/P(B)P(A\mid B)=P(A\cap B)/P(B) gives 1/2=(1/6)/P(B)1/2=(1/6)/P(B), so P(B)=1/3P(B)=1/3. This incorporates the Teacher completion of the Student blank.

How did this part go?

Common mistakes

A pair can be neither independent nor mutually exclusive. Test the two definitions separately.

Read the hint