Chapter 4 Probability and counting ruels

1. Chapter 4 Probability and counting ruels Math250 Southern University @ New Orleans

2. Section 4-1 Sample Spaces and Probability? Probability as a general concept can be defined as the chance of an event occurring. 1. If some event definitely will happen, we say the chance is 100% percent. Or say its probability for it to happen is 1. 2. If tossing a coin, then the chances to get head or tail is half-half. So we say the chance to get head is 50%. Or say that the probability to get head is 0.5. 3. If some event will definitely not happen, then we say its chance is zero. Or say its probability to happen is 0. 4. If the probability is greater than 0.5, we will say it is likely to happen. Otherwise, we will say it will be unlikely to happen.

3. A few basic concepts • A probability experiment: a chance process that leads to well-defined called outcomes. • An outcome is the result of a single trial of a probability experiment. • All possible different outcomes of a probability experiment is call a sample space. • An event is a set of outcomes of a probability experiment.

4. A formula for classical probability Classical probability assume that in the sample space all outcomes are equally likely to occur; suppose a sample space contains n outcomes, and an event E contains m outcomes, then its probability of the event E is

5. Some Probability Rules • Rule 1: The probability of any event E is a number between and including 0 and 1. 0 ≤ P(E) ≤ 1 • Rule 2: If an event E cannot occur, which means the event E is not in the sample space, then its probability is 0. P(E) = 0 • Rule 3: If an event E is certain occur, then its probability is 1. P(E) = 1

6. E

7. Sample space examples

8. Sample space for rolling two dice What is the probability that the sum of two dice number is 8?

9. Tossing a coin When tossing a coin, there are only two outcomes, head or tail. Because the chance to get head or tail is equal, so the probability to get head is 1/2 = 0.5

10. Tossing two coins When tossing two coins, what is the probability to get one head and one tail? To solve this question , we design outcomes into two events: either both same or different. We can feel that both events will have the equal chance to happen. Therefore, the probability to get one head and one tail is 50% or 0.5.

11. Tossing outcomes tree We try to use outcomes tree to solve the above question. We mark two coins as coin1 and coin2. Then put the outcomes of coin1 in the first line and the outcomes of the coin2 in the second line. Then we get the following outcomes tree H T coin1 coin2 HH HT TH TT There are 4 outcomes, so the probability is 2/4 = 0.5

12. Tossing three coins When tossing three coins, what is the probability to get two heads and one tail? We draw outcome tree coin1 H T coin2 T H T H coin3 H T H T H T H T HHH HHT HTH HTT THH THT TTH TTT There are 8 outcomes, so the probability is 3/8 = 0.375

13. Exercise When tossing four coins, what is the probability to get two heads and two tails? draw outcome tree first.

14. Rolling a die Ex 1: What is probability to get number 3 for rolling a die? Solution: Totally we have 6 outcomes and get number 3 is one of them. So the probability is 1/6 = 0.167. Ex 2: What is probability to get number less then 7 for rolling a die? Solution: All numbers of outcomes are less than 7. Therefore the probability to get number less then 7 is 1. Ex 3: What is probability to get number 9 for rolling a die? Solution: All number of a die is from 1 to 6. So we never could get 9. Therefore the probability to get number 9 is zero.

15. Drawing card A regular deck of cards has 52 cards of four types as the following

16. Drawing one card

17. Section 4-2 The Addition rules for probability B A

18. Mutually exclusive • Two events are mutually exclusive events if they cannot occur at the same time, i.e, they have no outcomes in common. B A

19. Addition rule 1 B A

20. Addition rule 2 The area in the intersection or overlapping part of both Circles corresponds to P(A and B) B A

21. Section 4-3 The Multiplication rules and conditional probability B A

22. Multiplication rule 1

23. An important case

24. Multiplication rule 2 • Dependent events: when the outcome or occurrence of the first event affects the outcome or occurrence of the second event in such a way that the probability changes. • When the two events are dependent, the probability of both occurring is

25. Conditional probability • The conditional probability of an event B in relationship to an event A is the probability that B occurs after event A has already occurred. • read as the probability of B given A.

26. Conditional Probability Explain the meaning of P(A), P(B), P(A|B) and P(B|A) using Vinn diagrams. A B A B

27. Re-solve two coins tossing When tossing two coins, what is the probability to get one head and one tail? Event B Event A

28. Re-solve three coins tossing When tossing three coins, what is the probability to get two heads and one tail?

29. Example The probability that Sam parks in a no-parking zone and get a parking ticket is 0.06. And the probability that Sam cannot find a legal parking lot and has to parking in a no-parking zone is 0.20. Today, Sam parks his car in a a no-parking zone. Find the probability that Sam will get a ticket.

30. Three coins tossing H T T H T H H T H T H T H T HHH HHT HTH HTT THH THT TTH TTT

31. Drawing Cards

33. Exercises