1. Ch 11.7 Probability

2. Definitions Experiment – any happening for which the result is uncertain Experiment – any happening for which the result is uncertain Outcome – the possible results of an experiment Outcome – the possible results of an experiment Sample Space – the set of all possible outcomes of the experiment Sample Space – the set of all possible outcomes of the experiment Event – any sub collection of a sample space Event – any sub collection of a sample space

3. Example 1 Find the sample space for each of the following Find the sample space for each of the following a. one coin is tossed {H, T} b. two coins are tossed {HH, HT, TH, TT} c. three coins are tossed

4. Probability If an event E has n(E) equally likely outcomes and its sample space S has n(S) equally likely outcomes, the probability of an event E is If an event E has n(E) equally likely outcomes and its sample space S has n(S) equally likely outcomes, the probability of an event E is P(E) = _n(E)_ n(S) n(S) Probability of an event is ALWAYS between 0 and 1 Probability of an event is ALWAYS between 0 and 1 If P(E) = 0, E is an impossible event If P(E) = 0, E is an impossible event If P(E) = 1, E is a certain event If P(E) = 1, E is a certain event

5. Monty Hall Problem In search of a new car, the player picks a door, say 1. The game host then opens one of the other doors, say 3, to reveal a goat and offers to let the player pick door 2 instead of door 1

6. Monty Hall Player's pick has a 1/3 chance while the other two doors have 1/3 chance each, for a combined 2/3 chance. Player's pick has a 1/3 chance while the other two doors have 1/3 chance each, for a combined 2/3 chance.

7. Monty Hall Player's pick remains a 1/3 chance, while the other two doors a combined 2/3 chance, 2/3 for the still unopened one and 0 for the one the host opened. Player's pick remains a 1/3 chance, while the other two doors a combined 2/3 chance, 2/3 for the still unopened one and 0 for the one the host opened.

8. Example 2 Two coins are tossed. What is the probability that both land heads up? Two coins are tossed. What is the probability that both land heads up? E = {HH} S = {HH, HT, TT, TH} P(E) = n(E) = 1 n(S) 4 n(S) 4

9. Example 3 A card is drawn from a standard deck of playing cards. What is the probability that is an ace? A card is drawn from a standard deck of playing cards. What is the probability that is an ace?

10. Example 4 Two six-sided dice are tossed. What is the probability the total of the dice is 7? Two six-sided dice are tossed. What is the probability the total of the dice is 7?

11. Example 5 Twelve-sided dice can be constructed such that each of the numbers from 1 to 6 appear twice on each die. Prove that these dice can be used in any game requiring ordinary six sided dice without changing the probability of different outcomes. Twelve-sided dice can be constructed such that each of the numbers from 1 to 6 appear twice on each die. Prove that these dice can be used in any game requiring ordinary six sided dice without changing the probability of different outcomes.

12. Example 6 In the Arizona state lottery, a player chooses six different numbers from 1 to 41. If these six numbers match the six numbers drawn, in any order, by the lottery commission, the player wins (or shares) the top prize. What is the probability of winning the top prize if the player buys one ticket? In the Arizona state lottery, a player chooses six different numbers from 1 to 41. If these six numbers match the six numbers drawn, in any order, by the lottery commission, the player wins (or shares) the top prize. What is the probability of winning the top prize if the player buys one ticket?

13. Example 7 The number of colleges and universities in various regions of the United States in 2003 is shown by a figure on pg 866. One institution is selected at random. What is the probability that the institution is in one of the three southern regions? The number of colleges and universities in various regions of the United States in 2003 is shown by a figure on pg 866. One institution is selected at random. What is the probability that the institution is in one of the three southern regions?

14. Mutually Exclusive Events Def – Two events A and B are mutually exclusive if A and B have no outcomes in common Def – Two events A and B are mutually exclusive if A and B have no outcomes in common Not Mutually Exclusive Not Mutually Exclusive

15. Example 8 One card is selected from a standard deck of 52 playing cards. What is the probability that the card is either a heart or a face card? One card is selected from a standard deck of 52 playing cards. What is the probability that the card is either a heart or a face card?

16. Example 8 The personnel department of a company has compiled data on the numbers of employees who have been with the company for various periods of time. The results are in the table. If an employees is chosen at random, what is the probability that the employee has The personnel department of a company has compiled data on the numbers of employees who have been with the company for various periods of time. The results are in the table. If an employees is chosen at random, what is the probability that the employee has a) 4 or fewer years b)9 or fewer years Years of Service # of employees 0-4157 5-989 10-1474 15-1963 20-2442 25-2938 30-3437 35-3921 40-448

17. Homework Pg 871 #1, 7-14,33,37,38,41,44,45 Pg 871 #1, 7-14,33,37,38,41,44,45