Probability

Basic Concepts of Probability and Counting

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.

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)

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?

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

#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

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

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

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 ≈

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.

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’)

Conditional Probability & the Multiplication Rule

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)

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)

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.

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: Nursing Majors Non-Nursing Majors TOTAL Males Females TOTAL

Ex continued… Find the probability that: 1. A randomly selected student is male. 2. A randomly selected student is a nursing major. 3. A randomly selected student is male, given that the student is a nursing major. 4. A randomly selected student is a nursing major, given that the student is male.

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)

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.