The Basics of Probability Theory Section 13.1 NCSCOS 2.02

Objectives: Determine sample space. Describe events as a subset. Compute Empirical Probability. Compute Theoretical Probability.

Key Terms: Experiment: any observation of a random phenomenon. Outcomes: the different possible results of the experiment. Sample space: the set of all possible outcomes for an experiment. Event: a subset of the sample space, denoted by E.

Example 1: Finding Sample Spaces We select an item from a production line and determine whether it is defective or not.

Example 2: Finding Sample Spaces Four children are born to a family and we note the birth order in respect to sex.

Some Good Advice: Although we usually describe events verbally, you should remember that an event is always a subset of the sample space. You can use the verbal description to identify the set of outcomes that make up the event.

Example 3: Describe events as subsets. A head occurs when we flip a coin.

Example 4: Describe events as subsets. Three girls and one boy are born to a family.

Key Terms: Probability of an Outcome (in a sample space): a number between 0 and 1 inclusive. Probability of an Event: defined as the sum of the probabilities of the outcomes that make up E, denoted by P(E).

Key Concept: Empirical Probability: if E is an event and we perform an experiment several times, then we estimate the probability of E as follows: P(E) = the number of times E occurs the number of times the experiment is performed

Example 5: Using Empirical Information to Assign Probabilities A pharmaceutical company is testing a new flu vaccine. The experiment is to inject a patient with the vaccine and observe the occurrence of side effects. Assume that we perform the experiment 100 times and obtain the information in the table. Based on the table, if a physician injects a patient with this vaccine, what is the probability that the patient will develop mild side effects? Side Effects Number of Times None67 Mild25 Severe8

Example 6: Using Empirical Probability We obtain red exactly once in two spins. Red Blue Yellow

Example 7: Using Empirical Probability Red appears exactly twice in three spins. Red Blue Yellow

Key Concept: Calculating Probability When Outcomes Are Equally Likely: If S is a sample space with all equally likely outcomes, then each outcome has a probability equal to: 1 = 1 number of outcomes in S n(S) For an event E in this sample space, P(E) = n(E) n(S)

Example 8: Computing Probability of Events What is the probability in a family with three children that two of the children are girls?

Example 9: Computing Probability of Events If we roll two fair dice, what is the probability of rolling a total of eight?

Example 10: Computing Probability of Events If we select two cards randomly from a standard 52- card deck, what is the probability that both are face cards?

Section 13.1 Assignment Classwork: TB pg. 730/1 – 20 All Remember you must write problems and show ALL work to receive credit for this assignment.

Section 13.1 Continued The Basic of Probability

Basic Properties of Probability Assume that S is a sample space for some experiment and E is an event in S < P(E) < 1 2. P(ø) = 0 3. P(S) = 1

Example 11: Using Probability to Explain Genetic Diseases TB pg. 731/25

Example 12: TB pg. 731/27

Example 13: TB pg. 731/29

Key Concept: Probability for Computing Odds. If E’ is the complement of the event E, then the odds against E are: P(E’) P(E)

Example 14: TB pg. 731/31

Example 15: TB pg. 731/33

Example 16: TB pg. 732/41

Example 17: TB pg. 732/

Example 18: TB pg. 733/51

Section 13.1 Assignment Class work: TB pg. 730/1 – 20 All…due 11/18 Remember you must write problems and show ALL work to receive credit for this assignment. TB pg. 731/26 – 52 Even…due 11/18 Remember you must write problems and show ALL work to receive credit for this assignment.