Conditional Probability

What

Conditional Probability is the probability of an event A occurring given that another event B has already occurred.

It is denoted as: \(P(A \mid B)\)

This expresses how the likelihood of A changes when we know B has happened.

How

The conditional probability is calculated using the formula:

If \(P(B) > 0\), then

\[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \]

Where:

• \(P(A \cap B)\) is the probability that both A and B occur.
• \(P(B)\) is the probability that B occurs (must be > 0).

Example:

If flipping two coins:

• Let A = first flip is heads

• Let B = both flips are heads

  Then,

\[ P(A \mid B) = \frac{P(\text{first = H and both = HH})}{P(\text{both = HH})} = \frac{P(HH)}{P(HH)} = 1 \]

Why

Conditional probability is essential because:

• It allows updating beliefs when new information is known (Bayesian reasoning).
• It helps model dependent events (where the outcome of one event affects another).
• It is foundational for concepts like Bayes’ Theorem, Markov Chains, and Machine Learning models.

It moves probability from static, one-time events to dynamic, context-aware inference.