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Probability

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

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Probability

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  1. 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. (c) 2007 IUPUI SPEA K300 (4392)

  2. 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 (c) 2007 IUPUI SPEA K300 (4392)

  3. 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 (c) 2007 IUPUI SPEA K300 (4392)

  4. 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 (c) 2007 IUPUI SPEA K300 (4392)

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

  6. 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? (c) 2007 IUPUI SPEA K300 (4392)

  7. 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 (c) 2007 IUPUI SPEA K300 (4392)

  8. 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. (c) 2007 IUPUI SPEA K300 (4392)

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

  10. 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 (c) 2007 IUPUI SPEA K300 (4392)

  11. 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 (c) 2007 IUPUI SPEA K300 (4392)

  12. 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? (c) 2007 IUPUI SPEA K300 (4392)

  13. 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. (c) 2007 IUPUI SPEA K300 (4392)

  14. 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 (c) 2007 IUPUI SPEA K300 (4392)

  15. 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? (c) 2007 IUPUI SPEA K300 (4392)

  16. 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? (c) 2007 IUPUI SPEA K300 (4392)

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

  18. Statistical Dependence 1 • Example 4-25, pp.206-207 • What if no replacement? • Suppose a blue ball is selected in the 1st trial • P(blue) is 2/10 in the 1st trial (c) 2007 IUPUI SPEA K300 (4392)

  19. 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. (c) 2007 IUPUI SPEA K300 (4392)

  20. 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 (c) 2007 IUPUI SPEA K300 (4392)

  21. 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) (c) 2007 IUPUI SPEA K300 (4392)

  22. 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 (2nd trial) and blue (1st 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 (2nd trial) and blue (1st trial) are dependent • Event blue in the 1st trial influences the probability of event white in the 2nd trial. (c) 2007 IUPUI SPEA K300 (4392)

  23. 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)=? (c) 2007 IUPUI SPEA K300 (4392)

  24. 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 (c) 2007 IUPUI SPEA K300 (4392)

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