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What is Conditional probability?

The probability that event A occurs given that event B has occurred. It is written P(A | B) and calculated as P(A | B) = P(A and B) ÷ P(B). It amounts to treating only the cases in which B occurred as the whole and considering the fraction of them in which A occurs. P(A | B) and P(B | A) are generally different values, and Bayes' theorem links the two. If A and B are independent, P(A | B) = P(A).

Japanese name: 条件付き確率

Conditional probability appears in 3 practice questions across 1 topic, including Probability, Statistics and Queueing.

Field: Probability, Statistics and Queueing

Try it: questions about Conditional probability

A fab processed 500 wafers in a month, 100 of them in chamber 2. A total of 40 wafers failed, and 15 of these had been processed in chamber 2. Which of the following is the conditional probability that a wafer processed in chamber 2 fails?

  1. A0.15
  2. B0.20
  3. C0.375
Show answer and explanation

Answer: A. 0.15

The conditional probability P(fail | chamber 2) treats only the wafers processed in chamber 2 as the whole and finds the fraction of them that failed: 15 ÷ 100 = 0.15. Written as a formula, P(fail | chamber 2) = P(fail and chamber 2) ÷ P(chamber 2) = (15/500) ÷ (100/500) = 0.15. It is a different value from P(chamber 2 | fail) = 15/40 = 0.375, with the condition and the outcome swapped, so care is needed about which one is asked for.

Conditional probability in the explanations

Questions that mention Conditional probability (3)

  1. A fab processed 500 wafers in a month, 100 of them in chamber 2. A total of 40 wafers failed, and 15 of these had been processed in chamber 2. Which of the following is the conditional probability that a wafer processed in chamber 2 fails?
  2. Among 10 lottery tickets, 3 are winners. Two tickets are drawn one after the other, and a drawn ticket is not put back. Which of the following is the probability that both are winners?
  3. The time to failure of a piece of equipment follows an exponential distribution with a mean of 1,000 hours. Take e^(−0.5) ≈ 0.61, e^(−1) ≈ 0.37 and e^(−1.5) ≈ 0.22. Given that the equipment has run for 500 hours without failure, which of the following is closest to the probability that it runs at least another 1,000 hours without failure?

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Related terms

Exponential distribution Bayes' theorem Binomial distribution Confidence interval Correlation coefficient Expected value Hypothesis testing Median Normal distribution Permutations and combinations

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