Conditional Probability
The conditional probability of an event A is the probability that event A will occur, given that some other event has occurred. Relative to the simple probability, the values assigned to the conditional probability of an event (given that another event occurred) is based on a sample space that is different than the one used to derive the simple probability; in fact, the sample space used to derive the conditional probability is the second or conditional event.
The conditional probability of event A given that event B occurred, also known as the conditional probability of A given B, is denoted by P(A|B) and is given by
where P (A Ç B) is the probability that both events A and B occur.
Two events A and B are independent if the occurrence of event A does not affect the probability that event B will occur, and vice-versa. In this case,
P(A|B) = P(A)
P(B|A) = P(B)
P(A Ç B) = P(A|B) P(B) = P(A) P(B)
The probability that two independent events occur together is simply the product of the probabilities for each independent event.
If events A and B are not independent, then they are dependent events.
EX. A card is drawn from a regular deck of 52 cards. Event A is the event that the card drawn is a Jack. Event B is the event that the card drawn is a diamond.
P(A) = 4 / 52 = 1 / 13 since there are 4 Jacks in the deck
P(B) = 13 / 52 = 1 / 4 since there are 13 diamonds in the deck
P(A Ç B) = 1 / 52 since there is only 1 Jack of diamonds in the deck
P(A|B) = P(A Ç B) / P(B) = (1 / 52) / (13 / 52) = 1 / 13 = P(A)
P(B|A) = P(A Ç B) / P(A) = (1 / 52) / (4 / 52) = 1 / 4 = P(B)
P(A Ç B) = P(A) P(B) = (1/13) (1/4) = 1/52
Events A and B are independent events. This is an intuitive thought, since each 13-card suit is a carbon copy of the other, save for the suit. Similarly, each card denomination (2 - 10,J,Q,K,A) has a club, diamond, heart and spade card.
