1. (c) 2007 IUPUI SPEA K300 (4392) Probability Likelihood (chance) that an event occurs Classical interpretation of probability: all outcomes in the sample space are equally likely to occur (random sampling) Empirical probability: conduct actual experiments to get the likelihood Subjective probability: ask professors, friends, mom, etc.

2. (c) 2007 IUPUI SPEA K300 (4392) Basic Concepts Probability experiment: a chance process that leads to well-defined results (outcomes) Outcome: the distinct possible result of a single trial of a probability experiment Sample space: the set of possible outcomes Event: identified with certain of outcomes

3. (c) 2007 IUPUI SPEA K300 (4392) Sample space Example 4-2 on page 180 Sample space: 52 outcomes Event “Queen”: 4 Event “Heart”: 12 Event “King Spade”: 1 Example 4-4 Sample space: 8 Event “Exactly two boys”: 3

4. (c) 2007 IUPUI SPEA K300 (4392) Tossing a Coin Tossing a coin once: head (H) or tail (T) Tossing two times: HH, HT, TH, TT Tossing three times: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT  2 X 2 X 2

5. (c) 2007 IUPUI SPEA K300 (4392) Tree Diagram H: head, T: tail

6. (c) 2007 IUPUI SPEA K300 (4392) Rolling a Die Rolling once: 1, 2, 3, 4, 5, 6 Rolling twice: (1, 1), (1,2)… (2, 1), (2, 2), …(6,6)  6^2 Rolling three times: (1,1,1), (1,1,2)… (1,2,1)… (1,6,6), (2,1,1)…(2,1,2)…(2,6,6), (3,1,1)…(6,6,6,)  6^3 Rolling four times: How to get the sample space?

7. (c) 2007 IUPUI SPEA K300 (4392) Combination Selecting r distinct objects out of n objects regardless of order at a time Example: select two students for awards among 5 students N factorial: n! = n X (n-1) X (n-2) X … 1 0! = 1

8. (c) 2007 IUPUI SPEA K300 (4392) Permutation An arrangement of n objects in a specific order using r objects at a time. Taking r ordered objects out of n objects at a time. Selecting one student for $10K award and another for $5K award among 5 students.

9. (c) 2007 IUPUI SPEA K300 (4392) Classical Probability P(E) is the probability that the event E occurs; expected (not actualized) likelihood The number of outcomes of event E, N E, divided by the number of total outcomes in the sample space, N.

10. (c) 2007 IUPUI SPEA K300 (4392) Probability Rules P(E) is a number between 0 and 1 Probability zero, P(E)=0, means the event will not occur. Probability 1, P(E)=1, means only the event occurs all the times. Sum of the probabilities of all outcomes in the sample space is 1

11. (c) 2007 IUPUI SPEA K300 (4392) Complementary Events the set of outcomes in the sample space that are not included in the outcome of E P(Ē) = 1 - P(E) P(E) = 1 - P(Ē) P(E) + P(Ē) = 1

12. (c) 2007 IUPUI SPEA K300 (4392) Empirical Probability Is your quarter really fair? Hmm… I guess the probability of head is larger than ½ for some reason. How about your die? Do all 1 through 6 have the equal chance of 1/6 to be selected? How can you check that?

13. (c) 2007 IUPUI SPEA K300 (4392) Addition Rule Probability that event A or B occurs P(A or B) = P(A) + P(B) – P (A and B) P(A U B) = P(A) + P(B) – P (A ∩ B) P(Nurse or Male)=P(N)+P(M)-P(N and M), Figure 4-5, p.198. Question 15, p.200.

14. (c) 2007 IUPUI SPEA K300 (4392) Mutually Exclusive Events P(A U B) = P(A) + P(B) - 0 P (A ∩ B) = 0 P(Monday or Sunday)=P(Monday)+P(Sunday)-0

15. (c) 2007 IUPUI SPEA K300 (4392) Multiplication Rule Probability that both events A and B occur P(A ∩ B) = P(A) X P(B) Example 4-24, p.206: P(queen and ace) = P(queen) X P(ace) = 4/52 X 4/52 Example 4-25: P(blue and white)=P(blue) X P(white) = 2/10 X 5/10 What if event A and B are related?

16. (c) 2007 IUPUI SPEA K300 (4392) Statistical Independence Occurrence of an event does not change the probability that other events occur. Occurrence of one measurement in a variable should be independent of the occurrence of others. Drawing a card with/without replacement. With replacement->independent (Ex. 4-25) Without replacement->dependent How do we know if two events are statistically independent?

17. (c) 2007 IUPUI SPEA K300 (4392) Examples How to put an elephant into a refrigerator? 1. Open the door 2. Put an elephant into the refrigerator 3. Close the door Now, how to put an hippo into the refrigerator? What makes a difference? Question 1, p.215

18. (c) 2007 IUPUI SPEA K300 (4392) Statistical Dependence 1 Example 4-25, pp.206-207 What if no replacement? Suppose a blue ball is selected in the 1 st trial P(blue) is 2/10 in the 1 st trial 1 st Trial2 nd Trial P(Blue)2/101/9 P(Red)3/103/9 P(White)5/105/9

19. (c) 2007 IUPUI SPEA K300 (4392) Statistical Dependence 2 P(blue then white)=2/10 X 5/10 w/o replacement P(blue then white)=2/10 X 5/9 w/ replacement 5/9: probability that event B (white ball) occurs given event A (blue) already occurred. Figure 4-6. p. 210. 1 st Trial2 nd Trial P(Blue)2/101/9 P(Red)3/103/9 P(White)5/105/9

20. (c) 2007 IUPUI SPEA K300 (4392) Conditional Probability P(B|A) is the probability that event B occurs after event A has already occurred. P(B|A)=P(A ∩ B) / P(A) P(A ∩ B)= P(A) X P(B|A) in case of statistical dependence

21. (c) 2007 IUPUI SPEA K300 (4392) Statistical Independence, again Events A and B are statistically independent, if and only If P(B|A)=P(B) or P(A|B)=P(A) Example 4-34, p.211 P(Yes|Female)=P(Female ∩ Yes) / P(Female) = [8/100]/[50/100]= 8/50 ≠ 40/100 Events Female and Yes are not independent P(A ∩ B)= P(A) X P(B|A)=[50/100]*[8/50]=8/100 P(A ∩ B)= P(A) X P(B) in case of statistical independence because P(B|A)=P(B) YesNoTotal Male321850 Female84250 4060100

22. (c) 2007 IUPUI SPEA K300 (4392) Examples: Example 4-25, p207 With Replacement: P(W|B)=P(W ∩ B)/P(B)= [2/10*5/10]/[2/10]=5/10=P(W) Events white (2 nd trial) and blue (1 st trial) are independent Without Replacement: P(W|B)=P(W ∩ B)/P(B)=[2/10*5/9]/[2/10] =5/10 ≠ P(W) Events white (2 nd trial) and blue (1 st trial) are dependent Event blue in the 1 st trial influences the probability of event white in the 2 nd trial.

23. (c) 2007 IUPUI SPEA K300 (4392) Examples: Question 34, p216 P(oppose|freshman)=[27/80]/[50/80]=27/50 P(sophomore|favor)=[23/80]/[38/80]=23/38 P(No opinion|sophomore)=? P(Favor | freshman)=? FavorOppose No opinion Total Freshman 1527850 Sophomore 235230 Total38321080

24. (c) 2007 IUPUI SPEA K300 (4392) Summary Addition: probability that event A or B occurs P(A U B) = P(A) + P(B) – P (A ∩ B) P (A ∩ B) =0 if mutually exclusive Multiplication: probability that both events A and B occur P(A ∩ B) = P(A) X P(B|A) P(B|A)=P(B) if statistically independent