Conditional Probability

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Conditional Probability: Definition, Formula, and Examples

What is conditional probability?

Conditional probability measures the chance that an event A occurs given that another event B has already occurred. It’s written P(A|B) and captures dependence: if the occurrence of B affects the likelihood of A, the events are dependent; if it doesn’t, they are independent.

Key relations
* Conditional probability: P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0.
* Chain rule (two events): P(A ∩ B) = P(A) · P(B|A) = P(B) · P(A|B).
* For multiple events: P(A ∩ B ∩ C) = P(A) · P(B|A) · P(C|A ∩ B) (or with appropriate conditional dependencies).

Formula and interpretation

P(B|A) = P(A ∩ B) / P(A)

This reads “the probability of B given A equals the probability that A and B both happen divided by the probability that A happens.” Intuitively, once you restrict the sample space to outcomes where A has occurred, P(B|A) is the fraction of those outcomes in which B also occurs.

Examples

  1. Marbles in a bag
  2. Bag: 6 red, 3 blue, 1 green (10 total).
  3. Event A: draw red. Event B: draw not green.
  4. P(B) = 9/10.
  5. P(A ∩ B) = P(A) = 6/10 = 3/5 (all red are not green).

  6. P(A|B) = (3/5) / (9/10) = 2/3.
    So, given the marble is not green, the chance it’s red is 2/3.

  7. Rolling a fair die

  8. Event A: roll even {2,4,6} (P(A)=1/2). Event B: roll >4 {5,6} (P(B)=1/3).
  9. A ∩ B = {6}, so P(A ∩ B)=1/6.

  10. P(B|A) = P(A ∩ B) / P(A) = (1/6) / (1/2) = 1/3.
    So, given the roll is even, the probability it is greater than 4 is 1/3.

  11. Sequential events (acceptance, scholarship, stipend)

  12. P(A) = 0.10 (accepted).
  13. P(B|A) = 0.02 (scholarship given acceptance).
  14. P(C|B) = 0.50 (stipend given scholarship).
  15. P(A ∩ B ∩ C) = 0.10 × 0.02 × 0.50 = 0.001 = 0.1%.
    This shows multiplying conditional probabilities to get the joint probability of a sequence of dependent events.

Conditional vs. marginal vs. joint probability

  • Marginal (unconditional) probability: P(A) — the chance of A regardless of other events.
  • Joint probability: P(A ∩ B) — the chance that A and B both occur.
  • Conditional probability: P(A|B) — the chance of A when you know B occurred.

Card example:
* Deck of 52 cards. A = draw a four (4/52 = 1/13). B = draw a red card (26/52 = 1/2).
* Joint P(A ∩ B) = 2/52 = 1/26 (four of hearts, four of diamonds).
* Conditional P(A|B) = (2/26) = 1/13 (given the card is red, chance it’s a four).

Bayes’ theorem (updating beliefs)

Bayes’ theorem reverses conditional probabilities and is central to updating probabilities with new evidence:

P(A|B) = [P(B|A) · P(A)] / P(B)

Use cases: medical testing, machine learning, finance — any situation where prior beliefs (P(A)) are updated by observed evidence (B) to produce a posterior probability P(A|B).

Simple intuition (Explain Like I’m Five)

If knowing that one thing happened makes another thing more or less likely, conditional probability tells you how much more or less likely. For example, if it was cloudy in the morning (B), the chance it will rain in the afternoon (A) changes compared with not knowing the morning weather.

Quick FAQs

  • What’s the difference between probability and conditional probability?
  • Probability (P(A)) considers A alone. Conditional probability (P(A|B)) considers A given that B happened.
  • What is prior probability?
  • The probability of an event before observing current data (used in Bayesian inference).
  • What is compound probability?
  • The probability of two independent events both occurring, found by multiplying their probabilities.

Bottom line

Conditional probability formalizes how new information changes the likelihood of events. It connects marginal and joint probabilities, enables sequential probability calculations, and underlies tools like Bayes’ theorem for updating beliefs when evidence arrives. Understanding conditional probability is essential for sound reasoning in statistics, data science, finance, and everyday decision-making.