1. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 14- 1

2. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 14 From Randomness to Probability

3. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 14- 3 Probability Thanks to the LLN, we know that relative frequencies settle down in the long run, so we can officially give the name probability to that value. Probabilities must be between 0 and 1, inclusive. A probability of 0 indicates impossibility. A probability of 1 indicates certainty.

4. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 14- 4 Equally Likely Outcomes When probability was first studied, a group of French mathematicians looked at games of chance in which all the possible outcomes were equally likely. It’s equally likely to get any one of six outcomes from the roll of a fair die. It’s equally likely to get heads or tails from the toss of a fair coin.

5. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 14- 5 Equally Likely Outcomes However, keep in mind that events are not always equally likely. A skilled basketball player has a better than 50-50 chance of making a free throw. When rolling a pair of dice, a sum of seven and a sum of twelve are not equally likely events.

6. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 14- 6 Formal Probability 1.Two requirements for a probability: A probability is a number between 0 and 1. For any event A, 0 ≤ P(A) ≤ 1.

7. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 14- 7 Formal Probability (cont.) 2.“Something has to happen rule”: The probability of the set of all possible outcomes of a trial must be 1. P(S) = 1 (S represents the set of all possible outcomes.)

8. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 14- 8 Formal Probability (cont.) Let an experiment consist of a collection of s mutually exclusive and equally likely events. This collection is called the sample space. Furthermore suppose exactly n of the events result in event A. Then the probability that event A will occur is P(A)=n/s.

9. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 14- 9 Formal Probability (cont.) An experiment consist of flipping a fair coin twice. Compute the probabilities of the following events. A - Exactly one head is observed. B - At least one head is observed. C - No tails are observed.

10. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 14- 10 Formal Probability (cont.) The first step is to construct the sample space. HHHTTHTT P(A) is 2/4 or 50%, P(B) is 3/4 or 75% and P(C) is 1/4 or 25%.

11. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 14- 11 Formal Probability (cont.) Repeat for three and four flips of a coin.

12. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 14- 12 Formal Probability (cont.) A pair of fair dice are rolled. Compute the probabilities of the following events. A - The sum of the two dice is 7. B - The sum of the two dice is 5 C - The sum of the two dice is an even number.

13. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 14- 13 Formal Probability (cont.) (1,1)(2,1)(3,1)(4,1)(5,1)(6,1) (1,2)(2,2)(3,2)(4,2)(5,2)(6,2) (1,3)(2,3)(3,3)(4,3)(5,3)(6,3) (1,4)(2,4)(3,4)(4,4)(5,4)(6,4) (1,5)(2,5)(3,5)(4,5)(5,5)(6,5) (1,6)(2,6)(3,6)(4,6)(5,6)(6,6)