Statistics and Probability

In probability theory, the concept of conditional probability is pivotal in determining the likelihood of an event given that another event has occurred. The formal definition is expressed by the formula \( P(A|B) = \frac{P(A \cap B)}{P(B)} \), where \( P(A|B) \) denotes the probability of event A occurring given that B has occurred, \( P(A \cap B) \) is the probability of both A and B occurring, and \( P(B) \) is the probability of event B. This relationship is crucial for understanding how the occurrence of one event affects the probability of another.

For independent events, where the occurrence of one event does not influence the occurrence of another, the conditional probability formula simplifies. If events A and B are independent, then \( P(A|B) = P(A) \) and \( P(A \cap B) = P(A) \times P(B) \), showing that the probability of A occurring given B has occurred is simply the probability of A occurring. The formulas are key in identifying whether events influence each other or can be considered independently.

Key Formulas:
Conditional Probability: \( P(A|B) = \frac{P(A \cap B)}{P(B)} \)
Independence: \( P(A|B) = P(A) \) and \( P(A \cap B) = P(A) \times P(B) \)

Formal Definition:

A and B are independent events if the occurrence of each one of them does not affect the probability that the other occurs.

This means that $$P(A \mid B) = P(A \mid B') = P(A)$$ and that $$P(B \mid A) = P(B \mid A') = P(B)$$.

Example:
Let's consider two events, A and B, within a standard deck of 52 playing cards. Event A is drawing an ace, and event B is drawing a heart. To find the probability of drawing an ace given that we have drawn a heart, we use the conditional probability formula: $$ P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{1}{52}}{\frac{13}{52}} = \frac{1}{13} $$ This result tells us that if we know a heart has been drawn, there is a 1 in 13 chance that it is also an ace.