Econ - Chapter 4
Multiplication law (Pg. 203)
*a probability law used to compute the probability of the intersection of two events* Whereas the addition law of probability is used to compute the probability of a union of two events, the multiplication law is used to compute the probability of the intersection of two events. The multiplication law is based on the definition of conditional probability P(A ∩ B) = P(B) * P(A | B) or P(A ∩ B) = P(A) * P(B | A)
Addition Law for Mutually Exclusive Events
P(A U B) = P(A) + P(B)
COMPUTING PROBABILITY USING THE COMPLEMENT
P(A) = 1 - P(A^c)
Permutations and Combinations
The counting rule for permutations closely relates to the one for combinations; however, an experiment results in more permutations than combinations for the same number of objects because every selection of n objects can be ordered in n! different ways.
Union of two events
The union of A and B is the event containing all sample points belonging to A or B or both. *The union is denoted by A U B.*
Venn Diagram
Venn diagrams illustrate the concept of a complement. The rectangular area represents the sample space for the experiment and as such contains all possible sample points. The circle represents event A and contains only the sample points that belong to A. The shaded region of the rectangle contains all sample points not in event A and is by definition the complement of A. In any probability application, either event A or its complement A^c must occur. Therefore, we have P(A) + P(A^c ) = 1
Two basic requirements for assigning probabilities
1. The probability assigned to each experimental outcome must be between 0 and 1, inclusively. 2. The sum of the probabilities for all the experimental outcomes must equal 1.0. Formulas - Pg. 184
Tree Diagram
A tree diagram is a graphical representation that helps in visualizing a multiple-step experiment. (A diagram used to show the sample space.)
Sample Point
An experimental outcome is also called a sample point to identify it as an element of the sample space.
Counting Rule for Multiple Step Experiments
If an experiment can be described as a sequence of k steps with n1 possible outcomes on the first step, n2 possible outcomes on the second step, and so on, then the total number of experimental outcomes is given by (n1) (n2) . . . (nk). S = {(H, H ), (H, T ), (T, H ), (T, T )}
Marginal Probabilities (Pg. 200)
Marginal probabilities are found by summing the joint probabilities in the corresponding row or column of the joint probability table. -Values found in the margins of joint probability tables (Usually totals)
Classical Method of Assigning Probabilities
The classical method of assigning probabilities is appropriate when all the experimental outcomes are equally likely. If n experimental outcomes are possible, a probability of 1/n is assigned to each experimental outcome. When using this approach, the two basic requirements for assigning probabilities are automatically satisfied.
Sample Space
The sample space for a random experiment is the set of all experimental outcomes. S = {1, 2, 3, 4, 5, 6}
Multiplication law for Independent Events
To compute the probability of the intersection of two independent events, we simply multiply the corresponding probabilities. Note that the multiplication law for independent events provides another way to determine whether A and B are independent. P(A ∩ B) = P(A) * P(B)
Mutually Exclusive Events
Two events are said to be mutually exclusive if the events have no sample points in common.
Subjective Method of Assigning Probabilities
is most appropriate when one cannot realistically assume that the experimental outcomes are equally likely and when little relevant data are available. When the subjective method is used to assign probabilities to the experimental outcomes, we may use any information available, such as our experience or intuition. After considering all available information, a probability value that expresses our degree of belief (on a scale from 0 to 1) that the experimental outcome will occur is specified. Because subjective probability expresses a person's degree of belief, it is personal. Using the subjective method, different people can be expected to assign different probabilities to the same experimental outcome. Example: Consider the case in which Tom and Judy Elsbernd make an offer to purchase a house. Two outcomes are possible: -E1 = their offer is accepted -E2 = their offer is rejected Judy believes that the probability their offer will be accepted is .8; thus, Judy would set P(E1) = .8 and P(E2) = .2. Tom, however, believes that the probability that their offer will be accepted is .6; hence, Tom would set P(E1) = .6 and P(E2) = .4. Note that Tom's probability estimate for E1 reflects a greater pessimism that their offer will be accepted. Both Judy and Tom assigned probabilities that satisfy the two basic requirements. The fact that their probability estimates are different emphasizes the personal nature of the subjective method.
Probability
likelihood that a particular event will occur
Joint Probabilities (Pg. 200)
the probability that two events occur together (probability of intersection of 2 events) ****Joint probability tables****
Random Experiments
A random experiment is a process that generates well-defined experimental outcomes. On any single repetition or trial, the outcome that occurs is determined completely by chance. Characteristics: -The experimental outcomes are well defined, and in many cases can even be listed prior to conducting the experiment. -On any single repetition or trial of the experiment, one and only one of the possible experimental outcomes will occur. -The experimental outcome that occurs on any trial is determined solely by chance.
Combinations
A second useful counting rule allows one to count the number of experimental outcomes when the experiment involves selecting n objects from a set of N objects. It is called the counting rule for combinations COUNTING RULE FOR COMBINATIONS - Pg. 183
Permutations
A third counting rule that is sometimes useful is the counting rule for permutations. It allows one to compute the number of experimental outcomes when n objects are to be selected from a set of N objects where the order of selection is important. The same n objects selected in a different order are considered a different ex peri mental outcome. COUNTING RULE FOR PERMUTATIONS - Pg. 183
Event
An event is a collection of sample points.
Complement of an Event
Given an event A, the complement of A is defined to be the event consisting of all sample points that are not in A The complement of A is denoted by A^c.
Intersection of two events
Given two events A and B, the intersection of A and B is the event containing the sample points belonging to both A and B. *The intersection is denoted by A ∩ B*
Probability of an Event
PROBABILITY OF AN EVENT - The probability of any event is equal to the sum of the probabilities of the sample points in the event. Using this definition, we calculate the probability of a particular event by adding the probabilities of the sample points (experimental outcomes) that make up the event *****maybe-(P(E) = number of favorable outcomes/total number of possible outcomes)
Independent Events
Two events A and B are independent if: P(A | B) = P(A) or P(B | A) = P(B) Otherwise, the events are dependent. However, if the probability of event A is not changed by the existence of event M—that is, P(A ∣ M) = P(A)—we would say that events A and M are independent events
Conditional Probability
Written P(A ∣ B). We use the notation ∣ to indicate that we are considering the probability of event A given the condition that event B has occurred. P(A | B) = P(A ∩ B) / P(B) or P(B | A) = P(A ∩ B) / P(A)
Addition Law
a probability law used to compute the probability of the union of two events P(A U B) = P(A) + P(B) - P(A ∩ B)
Relative Frequency Method of Assigning Probabilities
is appropriate when data is available to estimate the proportion of the time the experimental outcome will occur if the experiment is repeated a large number of times. | 0 | 1 | 2 | 3 | 4 | Number Waiting | 2 | 5 | 6 | 4 | 3 | Number of Days Outcome Occurred = Total 20