1. Section 6.2 Probability Models
Honors Statistics
November 12th, 2014
Mr. Calise

2. AP Statistics, Section 6.2, Part 1
Sample Space
The sample space S of random phenomenon is the set of all possible outcomes.
AP Statistics, Section 6.2, Part 1

3. AP Statistics, Section 6.2, Part 1
Event
An event is any outcome or a set of outcomes of a random phenomenon.
That is, an event is a subset of the sample space
AP Statistics, Section 6.2, Part 1

4. AP Statistics, Section 6.2, Part 1
Probability Model
A probability model is a mathematical description of a random phenomenon consisting of two parts: a sample space S and a way of assigning probabilities to events.
AP Statistics, Section 6.2, Part 1

5. 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·b number of ways.
AP Statistics, Section 6.2, Part 1

6. Sample Space: Rolling 2 Dice
1,1
1,2
1,3
1,4
1,5
1,6
2,1
2,2
2,3
2,4
2,5
2,6
3,1
3,2
3,3
3,4
3,5
3,6
4,1
4,2
4,3
4,4
4,5
4,6
5,1
5,2
5,3
5,4
5,5
5,6
6,1
6,2
6,3
6,4
6,5
6,6
AP Statistics, Section 6.2, Part 1

7. Sample Space: Flipping a Coin and Rolling a Die
AP Statistics, Section 6.2, Part 1

8. Sample Space: Flipping 3 Coins
TTT
TTH
THT
HTT
THH
HTH
HHT
HHH
AP Statistics, Section 6.2, Part 1

9. Sample Space: Flipping 4 Coins
TTTT
TTTH
TTHT
THTT
HTTT
TTHH
THTH
THHT
HTTH
HTHT
HHTT
THHH
HTHH
HHTH
HHHT
HHHH
AP Statistics, Section 6.2, Part 1

10. AP Statistics, Section 6.2, Part 1
Notation
AP Statistics, Section 6.2, Part 1

11. Probability Rules: Rule 1
AP Statistics, Section 6.2, Part 1

12. Probability Rules: Rule 2
AP Statistics, Section 6.2, Part 1

13. Probability Rules: Rule 3
AP Statistics, Section 6.2, Part 1

14. Probability Rules: Rule 4
AP Statistics, Section 6.2, Part 1

15. Probability Rules: Rule 4 (Different Notation)
AP Statistics, Section 6.2, Part 1

16. AP Statistics, Section 6.2, Part 1
Example
Marital status:
Never married
Married
Widowed
Divorced
Probability:
.298
.622
.005
.075
What is P(Married)?
P(Married)=.622
AP Statistics, Section 6.2, Part 1

17. AP Statistics, Section 6.2, Part 1
Example
Marital status:
Never married
Married
Widowed
Divorced
Probability:
.298
.622
.005
.075
What is P(not Married)?
P(not Married)= =.378 (Complement Rule)
AP Statistics, Section 6.2, Part 1

18. AP Statistics, Section 6.2, Part 1
Example
Marital status:
Never married
Married
Widowed
Divorced
Probability:
.298
.622
.005
.075
What is P(Never married or Divorced)?
Since “Never married and Divorced are disjoint, P(Never married or Divorced)= =.373 (Addition Rule for disjoint events)
AP Statistics, Section 6.2, Part 1

19. Probabilities in a finite space
Assign a probability to each individual outcome. These probabilities must be numbers between 0 and 1 and must have sum of 1.
The probabilities of any event is the sum of the probabilities of the outcomes making up the event.
AP Statistics, Section 6.2, Part 1

20. AP Statistics, Section 6.2, Part 1
Benford’s Law
Benford’s Law is the distribution of first digits in tax records, payment records, invoices, etc.
This distribution is handy in spotting illegitimate records.
AP Statistics, Section 6.2, Part 1

21. AP Statistics, Section 6.2, Part 1
Example
First Digit 1 2 3 4 5 6 7 8 9
Probability:
.301
.176
.125
.097
.079
.067
.058
.051
.046
AP Statistics, Section 6.2, Part 1

22. AP Statistics, Section 6.2, Part 1
Example
First Digit 1 2 3 4 5 6 7 8 9
Probability:
.301
.176
.125
.097
.079
.067
.058
.051
.046
AP Statistics, Section 6.2, Part 1

23. AP Statistics, Section 6.2, Part 1
Example
First Digit 1 2 3 4 5 6 7 8 9
Probability:
.301
.176
.125
.097
.079
.067
.058
.051
.046
AP Statistics, Section 6.2, Part 1

24. Equally Likely Outcomes
AP Statistics, Section 6.2, Part 1

25. Definition of Independence
Two events A and B are independent if knowing that one occurs does not change the probability of that the other occurs.
If A and B are independent,
This is the multiplication rule for independent events
AP Statistics, Section 6.2, Part 1

26. Example of Independent Events
First coin flip, second coin flip
Rolling of two dice
Choosing two cards with replacement
AP Statistics, Section 6.2, Part 1

27. Example of Not Independent Events
Choosing two cards without replacement
Scoring above 600 on verbal SAT, scoring 600 on math SAT
AP Statistics, Section 6.2, Part 1

28. Independent and complements
If A and B are independent, then so are…
Ac and Bc
A and Bc
Ac and B
AP Statistics, Section 6.2, Part 1

29. Are these events independent?
A={person is left-handed}
B={person is an only child}
C={person is blue eyed}
AP Statistics, Section 6.2, Part 1

30. Are these events independent?
A={person is college graduate}
B={person is older than 25}
C={person is a bank president}
AP Statistics, Section 6.2, Part 1

31. AP Statistics, Section 6.2, Part 1
Traffic light example
Suppose the timing of the lights on morning commute are independent.
The probability of being stopped at any light is .6
P(getting through all 6 lights)
.46=
P(getting stopped at all the lights)
.66=
AP Statistics, Section 6.2, Part 1

32. AP Statistics, Section 6.2, Part 1
Assignment
Exercises: , , odd
AP Statistics, Section 6.2, Part 1