1. Probability Models
Section 6.2

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.

3. Examples

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

5. The 36 possible outcomes in rolling two dice.

6. Sample Space for rolling two dice

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

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

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

11. 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.

12. Homework
Problems 11, 12, 17, 18

13. 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)

14. 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)

15. 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.

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

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

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

19. 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.

20. 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

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

22. Find probabilities First digit = P(A) = 0.301
P(B) = P(6) + P(7) + P(8) + P(9) =
= 0.222

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

24. 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

25. 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

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