Porisma

Worked solution

Find P(A) before testing independence

Events A and B satisfy P(B)=2/3P(B)=2/3, P(A∣B)=3/5P(A\mid B)=3/5 and P(A∩B′)=1/4P(A\cap B')=1/4. State with a reason whether A and B are independent.

CJC · Lecture · Self-Practice 3

Find the part of A inside B

P(A∩B)=P(A∣B)P(B)=(3/5)(2/3)=2/5P(A\cap B)=P(A\mid B)P(B)=(3/5)(2/3)=2/5.

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Combine both parts of A and compare

P(A)=P(A∩B)+P(A∩B′)=2/5+1/4=13/20P(A)=P(A\cap B)+P(A\cap B')=2/5+1/4=13/20. Since P(A∣B)=3/5=12/20≠13/20P(A\mid B)=3/5=12/20\ne13/20, A and B are not independent. This explains the printed answer No; the derivation is supplementary.

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Common mistakes

The given 1/4 is the probability of A and not B, not P(A). Add the two disjoint parts to recover P(A).

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