Probability Models Section 6.2

Probability Models The sample space S of a random phenomenon is the set of all possible outcomes. The event is any outcome or a set of outcomes of a random phenomenon. That is, an event is a subset of the sample space. A probability model is a mathematical description of a random phenomenon consisting of two: a sample space S and a way of assigning probabilities to events.

Examples

Sample space for rolling two-dice Make an outcome diagram for rolling two dice

The 36 possible outcomes in rolling two dice.

Sample Space for rolling two dice

Lets Practice: Provide a sample space for random digits from table B. Provide a sample space for flipping a coin and rolling a die.

Random Digit S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

Coin and Die S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6} Now make a tree diagram for the outcomes

Multiplication Principle If you can do one task in a number of ways and a second task in b number of ways, then both tasks can be done in a x b number of ways.

Homework Problems 11, 12, 17, 18

Probability Rules Rule 1: the probability P(A) of any event A is between 0 and 1 inclusive. Rule 2: If S is the sample space in a probability model, then P(S) = 1 Rule 3: the Complement of any event A is the event that A does not occur. Complement rule P(Ac) = 1 – P(A)

More rules Rule 4: Two events A and B are disjoint (mutually exclusive) if they have no outcomes in common. Addition rule for disjoint events P(A or B) = P(A) + P(B)

Independent Events Rule 5: P(A and B) = P(A)P(B) Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs.

Example Marital status: Never married 0.298 Married 0.622 Widowed 0.005 Divorced 0.075

What is the probability of a woman not being married? P(not married) = 1 – P(married) = 1 – 0.622 = 0.378

Which two events are disjoint? Never married and divorced P(never married or divorced) = P(never married) + P(divorced) = 0.298 + 0.075 = 0.373

Benford’s Law The first digits of numbers in legitimate records often follow a distribution known as Benford’s Law. These records are tax returns, payment records, invoices, expense account claims, etc.

Benford’s Law First digit Probability 1 0.301 2 0.176 3 0.125 4 0.097 0.079 6 0.067 7 0.058 8 0.051 9 0.046

Consider the events: A = {first digit is 1} B = {first digit is 6 or greater}

Find probabilities First digit = P(A) = 0.301 P(B) = P(6) + P(7) + P(8) + P(9) = 0.067+0.058+0.051+0.046 = 0.222

What about the probability that a digit is anything other than a 1? P(Ac) = 1 – P(A) = 0.699

Disjoint events What is the probability that the first digit is 1 or is 6 or greater? P (A or B) = P(A) + P(B) = 0.523

Random digits What is the probability that a randomly chosen first digit is 6 or greater? P(B) = 1/9 + 1/9 + 1/9 + 1/9 = 0.444

Assignment Problems 19, 22, 26, 28, 32, 34, 36, 39, 41 Due next class meeting.