6.2 Probability Models

An Event is an outcome or a set of outcomes of a random phenomenon An Event is an outcome or a set of outcomes of a random phenomenon. That is, an event is a subset of the sample space.

Ex.) Flip a coin three times HHH HHT HTT HTH TTT TTH THH THT “Exactly two heads” is an event A A = {HHT HTH THH}

In a probability model, events have the following properties: 1. Any probability is a number between 0 & 1 2. All possible outcomes together must have a probability 1. 3. The probability that an event does not occur is 1 minus the probability that the event does occur. 4. If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities. ex) Event 1 occurs 40% of the time Event 2 occurs 25% of the time and both can never occur together., then one or the other occurs 65% of the time because 40+25 = 65%

Probability Rules Rule 1: The probability P(A) of an event A satisfies 0<P(A)<1

Rule 2: If S is the sample space in a probability model, then P(S)=1

The COMPLEMENT RULE states P(A ) = 1 – P(A) Rule 3: The COMPLEMENT of any event A is the event that A does not occur, written as A . The COMPLEMENT RULE states P(A ) = 1 – P(A) c c

Rule 4: Two events A and B are DISJOINT if they have no outcomes in common and so can never occur simultaneously. If A and B are disjoint, P(A or B) = P(A) + P(B) This is the ADDITION RULE for disjoint events

Ex.) Select a woman age 25-29 at random Marital Status never married married widowed divorced Probability .353 .574 .002 .071. Each probability is between 0 & 1 All 4 possibilities add to 1 because these outcomes make the sample space S The probability that the woman we draw is not married is, by the complement rule P (not married) = 1 – P (married) 1 - .574 = .426 So 57.4% married and 42.6% not married. “never married” & “divorced” are disjoint P ( never married or divorced ) = P (never married) + P (divorced) = .353 + .071 = .424 So 42.4% of women are either never married or divorced

Probabilities in a Finite Sample Space Assign a probability to each individual outcome. These probabilities must be between 0 & 1 and sum to 1. The probability of any event is the sum of the probabilities of the outcomes making up the event.

Assigning Probabilities: Equally Likely Outcomes If a random phenomenon has K possible outcomes, all equally likely, then each individual outcome has probability 1/K. The probability of any event A is P(A)= Count of Outcomes in A Count of Outcomes in S = Count of Outcomes in A K

Most random phenomena do not have equally likely outcomes, so the general rule for finite sample spaces is more important than the special rule for equally likely outcomes

The Multiplication Rule for Independent Event Rule 5: Two events A and B are independent in knowing that one occurs does not change the probability that the other occurs. If A and B are independent . P(A and B) = P(A)P(B) This is the Multiplication Rule for independent events

Do not confuse disjoint with independent If A and B are independent, their complement A and B are independent c c