Conditional Probability

Conditional probability measures the probability that an event $ A $ occurs given that another event $ B $ has already occurred. It is denoted $ P(A|B) $, read as probability of A given B, and is calculated using the formula:

$$ P(A|B) = \frac{P(A \cap B)}{P(B)} $$

where $ P(A \cap B) $ is the probability that $ A $ and $ B $ occur simultaneously, and $ P(B) $ is the probability that $ B $ occurs (a probability that cannot be zero).

Intuitive interpretation: conditioning by $ B $ means restricting the set of possibilities to the single case where $ B $ is true, and then measuring the frequency of $ A $ in this new set.

Bayes' theorem reformulates a conditional probability by inverting the condition:

$$ P(B|A) = \frac{P(A|B) \cdot P(B)}{P(A)} $$

Bayesian inference allows us to calculate B given A from A given B, and vice versa.

Conditional probabilities are used to integrate available information into the calculation. They allow to:

— update a hypothesis in light of new data (medical tests, diagnoses, forecasts);

— model dependencies between events;

— avoid misinterpretations by distinguishing between correlation and causation.

The conditional probability $ P(A|B) $ is the probability of $ A $ occurring under the condition that $ B $ occurs.

The joint probability $ P(A\cap B) $ is the probability that $ A $ and $ B $ occur together, without any prior condition.

These two probabilities are related by the formula: $$ P(A|B) \cdot P(B) = P(A \cap B) $$

The two probabilities $ P(A|B) $ and $ P(B|A) $ are generally different and often mistakenly confused.

Two events $ A $ and $ B $ are independent if the occurrence of the first does not change the probability of the other, that is, if $ P(A|B) = P(A) $ or, equivalently, $ P(A \cap B) = P(A) \cdot P(B) $

This independence is symmetric and also implies $ P(B|A) = P(B) $