______ is the probability of Event Y occurring, given that Event Z has already occurred.

Question

BlackTom AI · Accepted Answer

Conditional probability P(Y|Z) denotes the probability that event Y occurs given that event Z has already occurred.

1. Relevant knowledge points and concepts
- Conditional probability: P(Y|Z) = P(Y ∩ Z) / P(Z), provided P(Z) > 0. It quantifies probability of Y in the restricted sample space where Z is true.
- Joint probability: P(Y ∩ Z) is the probability that both Y and Z occur together (no conditioning).
- Marginal probability: P(Y) or P(Z) is the overall probability of an event irrespective of other events.
- Posterior probability: in Bayesian context, the posterior is the updated probability of a hypothesis after observing data; formally it is a conditional probability but usually refers to parameter inference.

2. Step-by-step reasoning mapping the wording to a concept
- The phrase "probability of Event Y occurring, given that Event Z has already occurred" explicitly uses the word "given" (or the notion of conditioning).
- Conditioning corresponds exactly to conditional probability P(Y|Z).

3. Illustrative calculation with intermediate steps
- Use the definition: P(Y|Z) = P(Y ∩ Z) / P(Z).
- Example: if P(Y ∩ Z) = 0.20 and P(Z) = 0.50, then P(Y|Z) = 0.20 / 0.50 = 0.40.
- Intermediate step showed numerator (joint probability) divided by denominator (probability of the condition).

4. Why the correct answer is correct
- The correct answer, "Conditional Probability," matches the standard definition and formula. The condition "given that Z has already occurred" is exactly the conditioning operation expressed by P(Y|Z).

5. Why the other options are incorrect
- Joint Probability: P(Y ∩ Z) gives probability both happen, not the probability of Y given Z.
- Marginal Probability: P(Y) ignores the fact Z has occurred; it is unconditional.
- Posterior Probability: While mathematically a conditional probability in Bayesian inference, "posterior" specifically refers to updated beliefs about parameters/hypotheses after data; the plain phrase about events given another event is best described by "Conditional Probability."

6. Conclusion
- Therefore the correct choice is Conditional Probability, which matches the provided answer.

______ is the probability of Event Y occurring, given that Event Z has already occurred. 单项选择题

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