STATISTICS 6.0 Conditional Probabilities “Conditional Probabilities”

6.0 Conditional Probability Conditional Probability –Conditional Probability is defined as the probability of Event A knowing that Event B has already occur. –The term conditional probability is : P [A/B] (Event A knowing that Event B has already occur

6.0 Conditional Probability Conditional Probability –For example, what is the probability that the total of two dice will be greater than 8 given that the first die is a 6? This can be computed by considering only outcomes for which the first die is a 6. Then, determine the proportion of these outcomes that total more than 8. All the possible outcomes for two dice are shown below:

Independent vs. Dependent Events 1.Events A and B are said to be independent of each other if the occurrence of Event B has no effect on probability A: P[A/B] = P[A] 2.Events A and B are dependent, then Event A depend on Event B, or event B depend on event A. (Conditional Probability) 6.0 Conditional Probability

Multiplication Rule of Probabilities 1.Used to calculate the joint probability of 2 events. Probability of these events occurring at the same time. 2.Defined as the intersection of two events, key word: “and” Denote as P[A∩B] = P[A and B] 6.0 Conditional Probability

Multiplication Rule of Probabilities -For 2 independent events: P[A and B] = P[A]. P[B] = P[A] x P[B] -For 2 dependent events: P[A and B] = P[A/B]. P[B] = P[A/B] x P[B] 6.0 Conditional Probability

Mutually Exclusive Events 1.Two events are considered to be mutually exclusive if they cannot occur at the same time during the experiment 2.Event that can occur at the same time is not mutually exclusive ( Conditional probability ) 6.0 Conditional Probability

Additional Rule of Probabilities 1.Used to calculate the probability of the union events. Probability of either events would occur 2.Defined as the union of two events, key word: “or” Denote as P[AuB] = P[A or B] 6.0 Conditional Probability

Additional Rule of Probabilities -For 2 mutually exclusive events: P[A or B] = P[A] + P[B] -For 2 not mutually exclusive events: P[A or B] = P[A] + P[B] – P[A and B] 6.0 Conditional Probability

Bayes’ Theoram -Thomas Bayes ( ) develop mathematical rule: P[B/A] = P[B]. P[A/B] (P[B]. P[A/B] + P[B’]. P[A/B’]) 6.0 Conditional Probability