ProbabilityProbability

RulesRules

0 ≤ P(A) ≤ 1 for any event A.0 ≤ P(A) ≤ 1 for any event A. P(S) = 1P(S) = 1 Complement: P(AComplement: P(Acc) = 1 – P(A)) = 1 – P(A) Addition: If A and B are disjoint Addition: If A and B are disjoint

events, P(A or B) = P(A) + P(B).events, P(A or B) = P(A) + P(B). Multiplication: If A and B are Multiplication: If A and B are

independent events, independent events, P(A and B) = P(A)P(B).P(A and B) = P(A)P(B).

ReminderReminder

Disjoint – mutually exclusive (no Disjoint – mutually exclusive (no outcomes in common, never occur outcomes in common, never occur simultaneously, one happens then simultaneously, one happens then the other. the other.

Independent – knowing one outcome Independent – knowing one outcome doesn’t change the other outcome.doesn’t change the other outcome.

Joint ProbabilityJoint Probability

JOINT – (opposite of disjoint) mutually JOINT – (opposite of disjoint) mutually inclusive (some common outcomes, inclusive (some common outcomes, can occur simultaneously). The union can occur simultaneously). The union is less than the sum of the individual is less than the sum of the individual probabilities.probabilities.

P(A or B) = P(A) + P(B) – P(A and B).P(A or B) = P(A) + P(B) – P(A and B).

ExercisesExercises

6.27, 6.30, 6.31, 6.33, 6.36-6.38, 6.27, 6.30, 6.31, 6.33, 6.36-6.38, 6.46, 6.47, 6.52, 6.536.46, 6.47, 6.52, 6.53

Conditional ProbabilityConditional Probability

Probability changes if we know that Probability changes if we know that some other event has occurred.some other event has occurred.

New Notation: P(A|B) read New Notation: P(A|B) read “Probability of A given the “Probability of A given the information about the probability of information about the probability of B”B”

Multiplication RuleMultiplication Rule

The probability that both of two The probability that both of two events A and B happens together events A and B happens together

P(A and B) = P(A)P(B|A)P(A and B) = P(A)P(B|A)

Conditional is P(B|A) which is to say Conditional is P(B|A) which is to say that B occurs given that A occurs.that B occurs given that A occurs.

Conditional ProbabilityConditional Probability

When P(A) > 0, the conditional When P(A) > 0, the conditional probability of B given A is probability of B given A is

P(B|A) = P(A and B)/ P(A)P(B|A) = P(A and B)/ P(A)

Homework: 6.54 – 6.56Homework: 6.54 – 6.56

Extended Multiplication RulesExtended Multiplication Rules

Intersection – the event that all of the Intersection – the event that all of the events occur.events occur.

P(A and B and C) = P(A)P(B|A)P(C|A and B)P(A and B and C) = P(A)P(B|A)P(C|A and B)