Porisma

Worked solution

Recover a marginal probability from a conditional one

Events A and B satisfy P(A)=5/12P(A)=5/12, P(A∣B′)=7/12P(A\mid B')=7/12 and P(A∩B)=1/8P(A\cap B)=1/8. Find

(i) P(B)P(B),

(ii) P(A∣B)P(A\mid B),

(iii) P(B∣A)P(B\mid A),

(iv) P(A∪B)P(A\cup B).

State, with reasons, whether A and B are independent or mutually exclusive.

CJC · Discussion · Question 2

Answer

(i) 1/21/2; (ii) 1/41/4; (iii) 3/103/10; (iv) 19/2419/24. Neither independent nor mutually exclusive.

(i) Split A into two disjoint pieces

P(A)=P(A∩B′)+P(A∩B)P(A)=P(A\cap B')+P(A\cap B), so P(A∩B′)=5/12−1/8=7/24P(A\cap B')=5/12-1/8=7/24. By the given conditional probability, 7/12=P(A∩B′)/P(B′)=(7/24)/P(B′)7/12=P(A\cap B')/P(B')=(7/24)/P(B'). Hence P(B′)=1/2P(B')=1/2 and P(B)=1−P(B′)=1/2P(B)=1-P(B')=1/2.

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(ii) Condition on B

P(A∣B)=P(A∩B)/P(B)=(1/8)/(1/2)=1/4P(A\mid B)=P(A\cap B)/P(B)=(1/8)/(1/2)=1/4.

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(iii) Reverse the conditioning carefully

P(B∣A)=P(B∩A)/P(A)=(1/8)/(5/12)=3/10P(B\mid A)=P(B\cap A)/P(A)=(1/8)/(5/12)=3/10. The numerator is the same intersection, but the conditioning denominator changes.

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(iv) Use inclusion–exclusion

P(A∪B)=P(A)+P(B)−P(A∩B)=5/12+1/2−1/8=19/24P(A\cup B)=P(A)+P(B)-P(A\cap B)=5/12+1/2-1/8=19/24.

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Check independence and mutual exclusivity separately

P(A)P(B)=(5/12)(1/2)=5/24≠1/8=P(A∩B)P(A)P(B)=(5/12)(1/2)=5/24\ne1/8=P(A\cap B), so A and B are not independent. Also P(A∩B)=1/8≠0P(A\cap B)=1/8\ne0, so they are not mutually exclusive.

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Two overlapping event circles A and B inside a sample-space rectangle. Only A outside B is blue, reconstructing the source shaded A intersection B-complement. That probability is 7/24; the diagram is not to scale.
Two overlapping event circles A and B inside a sample-space rectangle. Only A outside B is blue, reconstructing the source shaded A intersection B-complement. That probability is 7/24; the diagram is not to scale.
Two overlapping event circles A and B inside a sample-space rectangle. Only A outside B is blue, reconstructing the source shaded A intersection B-complement. That probability is 7/24; the diagram is not to scale.

Common mistakes

A conditional probability is not generally the corresponding marginal; independence and mutual exclusivity are different conditions.

Solution notes

All printed asks, working, probability calculations and source annotations visually compared and retained. No marks are printed for this question.

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