Section 13.1 The Basics of Probability Theory 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?
Example 6: • Using Empirical Probability • We obtain red exactly once in two spins.
Example 7: • Using Empirical Probability • Red appears exactly twice in three spins.
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. • 0 <P(E)< 1 • P(ø) = 0 • 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. 732/47
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.