Conditional Probability
Conditional probability is a mathematical concept that measures the likelihood of an event occurring given that another event has already occurred. It quantifies the probability of an outcome based on additional information or circumstances.
Formula
The conditional probability of event A given event B is expressed as:
P(A | B) = P(A ∩ B) / P(B)
Where:
- P(A | B) represents the conditional probability of event A given event B
- P(A ∩ B) denotes the joint probability of both events A and B occurring simultaneously
- P(B) indicates the probability of event B occurring
Interpretation
The conditional probability measures the revised probability of event A occurring after event B has been observed. It takes into account the new information provided by the occurrence of event B.
Example
Suppose we have a deck of playing cards, including 52 cards in total. We want to calculate the probability of drawing a heart from the deck given that the card drawn is red. Let event A be drawing a heart, and event B be drawing a red card.
P(A | B) = P(A ∩ B) / P(B)
The probability of drawing a heart and a red card (A ∩ B) is 26 out of 52, since there are 26 red cards and half of them are hearts. The probability of drawing a red card (P(B)) is 26 out of 52 as well.
P(A | B) = (26/52) / (26/52) = 1/2
Therefore, the conditional probability of drawing a heart given that the card drawn is red is 1/2.