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Worked solution

Use a conditional to find three event probabilities

Given P(A)=0.7P(A)=0.7, P(B)=0.5P(B)=0.5 and P(A∣B)=0.8P(A\mid B)=0.8, find (i) P(A∩B)P(A\cap B), (ii) P(A∪B)P(A\cup B), (iii) P(B′∣A)P(B'\mid A).

[MJC/2004/II/30]

CJC · Lecture · Self-Practice 5

(i) Recover the intersection

P(A∩B)=P(A∣B)P(B)=0.8×0.5=0.4P(A\cap B)=P(A\mid B)P(B)=0.8\times0.5=0.4.

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(ii) Subtract the overlap once

P(A∪B)=P(A)+P(B)−P(A∩B)=0.7+0.5−0.4=0.8P(A\cup B)=P(A)+P(B)-P(A\cap B)=0.7+0.5-0.4=0.8.

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(iii) Condition on A after removing its overlap with B

P(A∩B′)=P(A)−P(A∩B)=0.7−0.4=0.3P(A\cap B')=P(A)-P(A\cap B)=0.7-0.4=0.3, so P(B′∣A)=0.3/0.7=3/7P(B'\mid A)=0.3/0.7=3/7. All printed answers 0.4,0.8,3/7 and the citation are retained; the derivations supplement them.

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Common mistakes

The given conditional is A given B; part (iii) asks for not B given A. Their denominators are different.

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