Conditional Probability
Conditional probability is a cncept in probability theory that describes the likelihood of an event occurring given that another event has already occurred. It's denoted as P(A|B), which reads as "the probability of A given B".
The formal definition of conditional probability is:
$P(A|B) = \frac{P(A \cap B)}{P(B)}$
Where:
- P(A|B) is the conditional probability of A given B
- P(A ∩ B) is the probability of both A and B occurring
- P(B) is the probability of B occurring
Let's consider a deck of 52 cards. We want to find the probability of drawing a king given that we've drawn a face card.
- P(King) = 4/52 = 1/13
- P(Face card) = 12/52 = 3/13
- P(King and Face card) = 4/52 = 1/13
P(King | Face card) = P(King and Face card) / P(Face card) = (1/13) / (3/13) = 1/3
So, the probability of drawing a king, given that we've drawn a face card, is 1/3.
NoteThe conditional probability formula can also be rearranged to: P(A ∩ B) = P(B) * P(A|B) This form is particularly useful when we know the conditional probability and want to find the probability of both events occurring.
Independent Events
Two events A and B are considered independent if the occurrence of one event does not affect the probability of the other event occurring. Mathematically, this is expressed as:
$P(A|B) = P(A)$
