CHAPTER THREE

1. CHAPTER THREE Probability

2. Section 3.1 Basic Concepts of Probability and Counting

3. Probability Experiment: • … an action, or trial, through which specific results are obtained. • The result of a single trial is called an OUTCOME. • The set of all possible outcomes is called the SAMPLE SPACE. • An EVENT is a subset of the sample space.

4. EX: Identify the sample space and determine the # of outcomes • Guessing a student’s letter grade (A, B, C, D, F) in a class. • Tossing three coins. (Hint… draw a tree diagram)

5. The Fundamental Counting Principle • If one event can occur in m ways and a second event can occur in n ways, the number of ways the two events can occur in sequence is m · n • EX: For dinner you select one each from 3 appetizers, 4 entrees, and 2 desserts. How many different ‘meals’ can you make if you choose one from each category?

6. 3 Types of Probability #1 Classical Probability (AKA Theoretical Probability): used when each outcome in a sample space is equally likely to occur. P(E) = probability of event E P(E) = # of outcomes in event E Total # of outcomes in sample space

7. #2 Empirical Probability (AKA Statistical Probability) Based on observations obtained from probability experiments. Same as relative frequency of event. P(E) = Frequency of Event E = f Total Frequency n

8. #3 Subjective Probability • Result from intuition, educated guesses, and estimates.

9. The Law of Large Numbers • As an experiment is repeated over and over, the empirical probability of an event approaches the theoretical probability of the event.

10. Classify as an example of classical, empirical, or subjective probability. • The probability of choosing 6 numbers from 1 to 40 that matches the 6 numbers drawn by a state lottery is 1/3,838,380 ≈ 0.00000026.

11. Rules of Probability 0 < P(E) < 1 The probability of an event is between 0 and 1 P(E) = 0 means the event CANNOT occur. P(E) = 1 means the event is CERTAIN. ΣP(E) = 1 The sum of the probabilities of all outcomes in the sample space is one.

12. Complementary Events • The complement of event E (denoted E’) is the set of all outcomes in the sample space that are NOT part of event E. • P(E) + P(E’) = 1 • P(E’) = 1 – P(E) • P(E) = 1 - P(E’)

13. Section 3.2 Conditional Probability & the Multiplication Rule

14. Conditional Probability … the probability of an event occurring, GIVEN that another event has occurred. The conditional probability of event B occurring given that event A occurred is P(B | A)

15. Independent & Dependent Events • Two events are INDEPENDENT if the occurrence of one does not affect the probability of the other event. • A and B are independent if… P(B | A) = P(B)

16. Dependent or Independent? • Returning a rented movie after the due date and receiving a late fee. • A ball numbered 1 through 52 is selected from a bin, replaced, and a second numbered ball is selected from the bin.

17. Ex (from p 152) • #7 The table shows the number of male and female students enrolled in nursing at the University of Oklahoma Health Sciences Center for a recent semester:

18. Ex continued… Find the probability that: • A randomly selected student is male. • A randomly selected student is a nursing major. • A randomly selected student is male, given that the student is a nursing major. • A randomly selected student is a nursing major, given that the student is male.

19. The Multiplication Rule • The probability that A and B will occur in sequence is: P(A and B) = P(A) · P(B | A) • If A and B are independent, use: P(A and B) = P(A) · P(B)

20. EX from p 154 • #27. The probability that a person in the US has type B+ blood is 9%. Five unrelated people in the US are selected at random. • A. Find the probability that all five have type B+ blood. • B. Find the probability that non of the five have type B+ blood. • C. Find the probability that at least one of the five has type B+ blood. • D. Which of the events can be considered unusual. Explain.