Exercise · ZW-CONDITIONAL-PROBABILITY-009 · ★★★ Advanced
With \(P(B)>0\), which statement is true?
Choose one option.
Show answer
- D. \(P(A\mid B)+P(A^c\mid B)=1\)
Explanation
Given \(B\), \(A\) and \(A^c\) split the whole reduced space, so their conditional probabilities add to 1. The others fail: conditioning can raise or lower a probability and \(P(A\mid B)\ne P(B\mid A)\) in general.
Review the concept: Conditional probability
The conditional probability of \(A\) given \(B\), \(P(A\mid B)\), is the probability that \(A\) occurs when \(B\) is already known to have occurred; it is computed within the space reduced to \(B\).
\[P(A\mid B)=\frac{P(A\cap B)}{P(B)},\qquad P(B)>0\]
More practice problems: conditional probability
- ZW-CONDITIONAL-PROBABILITY-001 If P(A∩ B)=0.2 and P(B)=0.5, what is P(A| B)?
- ZW-CONDITIONAL-PROBABILITY-002 A fair die is rolled and it is known the result is even. What is the probability that it is 6?
- ZW-CONDITIONAL-PROBABILITY-003 Two fair dice are rolled and the first shows 3. What is the probability that the sum is 8?
- ZW-CONDITIONAL-PROBABILITY-004 If P(A)=0.6, P(B)=0.4 and P(A∪ B)=0.7, what is P(A| B)?
- ZW-CONDITIONAL-PROBABILITY-005 An urn has 5 red and 3 blue balls. Two are drawn without replacement. Given that the first was red,…
- ZW-CONDITIONAL-PROBABILITY-006 Two cards are drawn from a 52-card deck without replacement. What is the probability that both are…
- ZW-CONDITIONAL-PROBABILITY-007 A family has two children and at least one is a girl. Assuming each child is equally likely to be a…
- ZW-CONDITIONAL-PROBABILITY-008 If P(A)=0.4 and P(B| A)=0.5, what is P(A∩ B)?
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