What is conditional probability (short answer)
– Conditional probability quantifies the chance of one event occurring when we know another related event has occurred. It is written P(A|B), read “the probability of A given B.”
Core definition and formula
– Let A and B be two events in the same probability space. The conditional probability of A given B is
P(A|B) = P(A ∩ B) / P(B),
provided P(B) > 0.
– Here P(A ∩ B) is the joint probability that both A and B occur. P(B) is the marginal (or unconditional) probability of B.
Key terms (defined)
– Conditional probability: Probability of an event with extra information that another event happened.
– Marginal (unconditional) probability: The probability of a single event ignoring any other information.
– Joint probability: The probability that two (or more) events happen together, written P(A ∩ B).
– Independent events: Two events A and B are independent if knowing one occurred does not change the probability of the other; formally P(A|B) = P(A).
– Prior probability: A probability assigned before observing new evidence (often P(A) in Bayesian updates).
– Compound probability: Probability of a sequence or combination of events (can involve conditional probabilities).
– Bayes’ theorem (relation): P(A|B) = P(B|A)·P(A) / P(B), useful for reversing conditioning.
Short checklist — how to compute a conditional probability
1. Define the sample space and count outcomes (or use probabilities if given).
2. Clearly define events A and B.
3. Compute P(B) (ensure P(B) > 0).
4. Compute P(A ∩ B) (the probability both events occur).
5. Apply P(A|B) = P(A ∩ B) / P(B).
6. Interpret the result and check it lies between 0 and 1.
Worked example — marbles in a bag
Scenario: A bag contains 6 red, 3 blue, and 1 green marble. Total marbles = 10.
Goal: Probability the drawn marble is red (event A) given it is not green (event B).
Step 1 — identify events:
– A = “draw a red marble.”
– B = “draw a marble that is not green.”
Step 2 — compute P(B):
– Non-green marbles = red + blue = 6 + 3 = 9.
– P(B) = 9 / 10.
Step 3 — compute P(A ∩ B):
– A ∩ B is simply “red and not green,” which equals “red.” So P(A ∩ B) = number of red / total = 6 / 10 = 3 / 5.
Step 4 — apply the formula:
– P(A|B) = P(A ∩ B) / P(B) = (3/5) / (9/10) = (3/5) × (10/9) = 2/3 ≈ 0.6667.
Interpretation: Given the marble drawn is not green, there is a 2/3 chance it is red.
Practical notes and checks
– Always ensure the conditioning event has positive probability (P(B) > 0). Otherwise P(A|B) is undefined.
– If A and B are independent, P(A|B) = P(A) and P(A ∩ B) = P(A)·P(B).
– Use Bayes’ theorem when you need P(A|B) but you know P(B|A), P(A), and P(B).
– In applied fields (finance, insurance, medicine), conditional probabilities model how new information updates risk estimates; assumptions about independence and the underlying probability model matter.
Common pitfalls
– Confusing P(A|B) with P(B|A). They are generally different.
– Forgetting to restrict the sample space to B when interpreting conditional results.
– Using empirical frequencies without verifying data representativeness.
Further reading (reputable resources)
– Investopedia — Conditional Probability: https://www.investopedia.com/terms/c/conditional_probability.asp
– Khan Academy — Conditional probability and independence (learning resource): https://www.khanacademy.org/math/statistics-probability/probability-library/probability-multiplication
– Wikipedia — Conditional probability: https://en.wikipedia.org/wiki/Conditional_probability
– StatTrek — Conditional Probability tutorial: https://stattrek.com/probability/conditional-probability.aspx
Educational disclaimer
This explainer is for educational purposes only and does not constitute individualized investment, legal, or professional advice.