Understanding Probability: Three Views and Basic Concepts

1. Chapter 7 Probability I Three Views of Probability A. Subjective-Personalistic View 1. Probability of event A, p(A), is a measure of the strength of one’s expectation that event A will or will not occur.

2. B. Classical or Logical View 1. Probability of event A is the number of events favoring A, nA, divided by the total number of equally likely events, nS, p(A) = nA/nS 2. Probability is based on a logical analysis. 3. p(A) is always a number between 0 and 1 because nA ≤ nS.

3. C. Empirical Relative-Frequency View 1. Probability of event A is anumber approached by the ratio nA/n as the total number of observations, n, approaches infinity. 2. If a head is obtained 12 times in 20 tosses, the best estimate of the probability of heads is nA/n = 12/20 = .6. If a head is obtained 120 times in 200 tosses, our confidence in the estimate 120/200 = .6 is even greater.

4. II Basic Concepts A. Experiment B. Compound and Simple Events 1. Compound events in tossing a die Event A—observe an odd number  Event B—observe an even number  Event C—observe a number less than 4

5. 2. Simple events in tossing a die Event E1—observe a 1 Event E2—observe a 2 Event E3—observe a 3 Event E4—observe a 4 Event E5—observe a 5 Event E6—observe a 6

6. III Graphing Simple and Compound Events A. Euler Diagram 1. Sample space, S, with nS sample points, Ei

7. B. Formal Properties of Probability 1. 0 ≤ p(Ei) ≤ 1 for all i 3. p(S) = 1 IV Probability of Combined Events A. Union of Events A and B: Set of Elements that Belongs to A or B or to Both A and B B. Intersection of Events A and B: Set of Elements that Belongs to Both A and B

8. V Addition Rule of Probability A. p(AorC) = p(A) + p(C) – p(AandC) 1. p(AorC) = 3/6 + 3/6 –2/6 = 2/3

9. B. Addition Rule of Probability for Mutually Exclusive Events where p(AandB) = 0 1. p(AorB) = p(A) + p(B) 2. p(AorB) = 3/6 + 3/6 = 1 3. Collectively exhaustive events: probability of union equals 1

10. VI Multiplication Rule of Probability A. Conditional Probability 1. p(A | C) = p(A and C)/p(C) 2. p(C | A) = p(A and C)/p(A)

11. 3. Example of conditional probability for events A and C.

12. B. Multiplication Rule of Probability 1. p(A and C) = p(A) p(C | A) = p(C) p(A | C) 2. p(A and C) = (3/6)(2/3) = (3/6) (2/3) = 1/3

13. C. Statistical Independence 1. Events A and C are independent if p(A | C) = p(A) 2. Events A and C are not independent because p(A | C) ≠ p(A) 2/3 ≠ 3/6

14. 3. Events A (obtain a H on the toss of a coin) and B (obtain a 5 on the roll a die) are statistically independent because p(B|A) = p(B).

15. D. Multiplication Rule for Statistically Independent Events where p(B|A) = p(B) 1. p(A and B) = p(A) p(B) 2. p(A and B) = (6/12)(2/12) = 1/12

16. VII Counting Simple Events A. Fundamental Counting Rule 1. If an event can occur in n1 ways and a second event in n2 ways and each ofthe first event’s n1 ways can be followed by any of the second’s n2 ways, then the number of ways thatevent 1 followed by event 2 can occur is n1 ×n2.

17. 2. Rolling a die (event 1) and tossing a coin (event 2) n1 n2 = (6)(2) = 12 3. k events, say rolling k = 3 dice n1 n2, . . . , nk = (6)(6)(6) = 216 B. Permutation of n Objects Taken n at a Time (nPn) 1. n factorial, n! =n(n– 1)(n– 2) . . . (1) 2. nPn = n! =n(n– 1)(n– 2) . . . (1)

18. 3. Consider n objects and a box with n compartments − − 4. The first compartment can be filled with any of the n objects, the second with any of n – 1 remaining objects, and the nth compartment with the one remaining object. 5. According to the fundamental counting rule nPn = n(n– 1)(n– 2) . . . (1)

19. 6. For n = 5 objects 5P5 = (5)(4)(3)(2)(1) = 120 C. Permutation of n Objects Taken r at a Time (nPr) 1. Consider n = 5 objects and a box with r = 3 compartments − − −

20. 2. The number of ways that n = 5 objects can be placed in r = 3 ordered compartments is given by 3. Alternative formula

21. D. Combination of n Objects Taken r at a Time (nCr) 1. nCr ignores the order of the objects by dividing nPr by the number of ways that r elements can be arranged which is r!.

22. 2. The number of ways that n = 5 objects can be placed in r = 3 compartments ignoring the order of the objects is given by