Read the event and condition before calculating
These original questions specify whether the quote is for or against an event, whether both event probabilities are positive, and which event supplies a conditional denominator. Keep those conditions attached to the formula.
Use a consistency check
Converted event and complement probabilities should sum to one. A joint probability cannot exceed either marginal. A conditional probability must use a positive conditioning probability. These checks help diagnose a result without replacing the reasoning.
Try the questions
Events A and B are mutually exclusive, with P(A) = 30% and P(B) = 20%. Are they independent?
- A.
Yes, because they do not overlap.
- B.
No, because zero does not equal 30%×20%.
- C.
The probabilities must sum to one before the relationship can be assessed.
Answer and explanation
Answer: B
Their joint probability is zero, while the product of the positive marginals is 6%. They do not satisfy independence.
- A
Non-overlap makes their joint probability zero; independence instead requires the product of the marginals.
- B
Both marginal probabilities are positive, so their product is positive.
- C
A pair of mutually exclusive events need not exhaust the full model.
P(A|B) = 65%, P(B) = 40% and P(A) = 50%. What is P(B|A)?
- A.
65%.
- B.
26%.
- C.
52%.
Answer and explanation
Answer: C
The overlap is 65%×40% = 26%; dividing by the new conditioning probability of 50% gives 52%.
- A
This reuses the conditional in the opposite direction without changing its denominator.
- B
This is the joint probability, before dividing by P(A).
- C
The overlap is measured relative to the A event for P(B|A).
A fictional model quotes odds of 9 to 1 against default. What default probability follows from that quote?
- A.
10%.
- B.
90%.
- C.
9%.
Answer and explanation
Answer: A
Default has favourable weight 1 out of a combined total of 10.
- A
Against default assigns the larger weight to the complement.
- B
This is the probability of the complement, no default.
- C
Dividing nine by one hundred is not the supplied odds normalization.