- CFA Exams
- CFA Level I Exam
- Study Session 2. Quantitative Methods (1)
- Reading 8. Probability Concepts
- Subject 3. Addition Rule for Probabilities: the Probability that at Least One of Two Events Will Occur

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**CFA Practice Question**

Of letters sent through a certain post office, 20% have insufficient postage, 48% have the wrong zip code, and 58% have one or both of these problems. What is the probability that a letter has both insufficient postage and the wrong zip code?

A. 0.10

B. 0.30

C. 0.68

**Explanation:**Let A denote the event that the letter has insufficient postage and B denote the event that it has the wrong zip code. We know that P(A) = 0.20, P(B) = 0.48 and P(A or B) = 0.58. Solving using the general rule of addition for P(A and B), one obtains P(A and B) = P(A) + P(B) - P(A or B). Thus, the probability of a letter having both problems is P(A and B) = 0.20 + 0.48 - 0.58 = 0.10.

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**User Contributed Comments**
4

User |
Comment |
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nostalgia |
Can anyone clarify this? If 58% have either one OR BOTH these problems, it doesn't sound like P(A or B) = 58%. It should be P(A or B) + P(AB) = 58%. |

kamcooler |
Think like this: Only A + Only B + Both = 58 Only A = 20 - Both Only B = 48 - Both Substitute into original: (20 - Both) + (48 - Both) + Both = 58 Solve for Both... |

linzlinked |
The questions is asking P(A and B)=?, but the equation above for P(AandB)=0.2+0.48-0.58=0.1 is for P (A or B), isn't it? I thought P (A and B) = P(A)*P(B) for 2 independent probabilities. Anybody can help clarify a bit? Thanks |

tferik |
My short cut thinking -If 20% has insufficient postage -The question asks for both insufficient postage and wrong zip -Therefore, the answer must be less than 20% because not all insufficient postage will have an incorrect zip -Of course, if there are other answer choices on the real exam, then, further work will be needed, but at least it can help eliminate a choice |