Chapter 6 Probability: The Study of Randomness Section 6.1 The Idea of Probability—the branch of math that describes the pattern of chance behavior. I. History of Probability a. As old as civilization itself. Archaeologists have found evidence that Egyptians used a small bone in mammals, the astralagus, as a sort of 4-sided die as early as 3500 B.C. b. Games of chance were common in Greek and Roman times and during the Renaissance of Western Europe c. Probability is thought of as beginning with the correspondence between Blaise Pascal and Pierre de Fermat in 1654 involving some problems related to gambling. d. In the 18th and 19th century repeated measurements in astronomy and surveying led to further advances in probability since repeated measurements are random and can be described by distributions. e. Today we use probability to study the flow of traffic, path of hurricanes, energy states of subatomic particles, insurance rates, genetics, medical diagnosis, etc.
II. Randomness a. Not a synonym for haphazard b. Individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions. Pg 332 Example 6.2 c. For random phenomenon the probability of any outcome is the proportion of times the outcome would occur in a very long series of repetitions. d. The probability is the long-term relative frequency.
Section 6.2 Probability Models I. Vocabulary of Probability Models a. Random phenomena or chance experiment— any activity or situation in which there is uncertainty about which of two or more possible outcomes will result. b. Sample space—the set of all possible outcomes of a random phenomena c. Event—any outcome or a set of outcomes of a random phenomena. The event is a subset of the sample space.
II. Ways to Describe Sample Space a. Making a Table Example 6.3 Events could be the following: A=event that the sum is 5 called ―Roll a 5‖ A= B=event that the sum is 7, a win in Craps B= C=event that the sum is 12, a loss in Craps C= b. Using a Tree Diagram Example 6.5 A=event that you roll a six A= B=event that you get a head B= c. Multiplication Principle—if one task is done in ―a‖ ways and a second task is done in ―b‖ ways, then both tasks can be done in a x b ways
Homework: 6.12-6.26 even
III. Forming New Events a. The complement of A, denoted, Ac, consists of all experimental outcomes that are not in event A.
b. The union of two events A or B, denoted A U B, consists of all outcomes in A or in B or in both
c. The intersection of two events A and B , denoted A ∩ B, consists of all outcomes that are in both A and B. d. Example: An observer stands at the bottom of a freeway off-ramp and observes the turning direction (L=left or R-right) of each of three successive vehicles. What is the sample space? A=event that exactly one of the cars turn right B=event that at most one of the cars turns right C=event that all cars turn in the same direction
D=event that not all cars turn in the same direction.
E=event that exactly one of the cars turns right or all cars turn in the same direction.
F=event that at most one car turns right and all cars turn in the same direction.
IV. Probability Rules P(A) = count of outcomes in A count of outcomes in S a. Any probability is a number between 0 and 1. 0 ≤ P(A) ≤ 1 b. All possible outcomes together must have probability 1. P(S) = 1 where S is the sample space c. The probability that an event does not occur is 1 minus the probability that the event does occur
P(Ac) = 1 – P(A) d. If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities. P(A or B) = P(A) + P(B) Two events that have no common outcomes are said to be disjoint or mutually exclusive Examples 6.8 – 6.9 pg 344