Decision Chapter 2 intro To Probability

HW Question Suppose that we have a sample space with five equally likely experimental outcomes: E1, E2, E3, E4, E5. Let A={E2, E4} B={E1, E3} C={E1, E4, E5}. A. Find P(A), P(B), and P(C). B.Find P(A ∪ B). C.Find A^C. Find C^C. (Enter your answer in set notation.) Find P(A^C) and P(C^C). D.Find A ∪ BC. (Enter your answer in set notation.) Find P(A ∪ BC) E.Find P(B ∪ C).

A. P(A)=.4 P(B)=.4 P(C)=.6 B. .8 C. {E1,E3,E5} {E2,E3,E5} P(A^C)=.6 P(C^C)=.4 D. {E2,E4,E5} E. .8

Let A be an event that a person's primary method of transportation to and from work is an automobile and B be an event that a person's primary method of transportation to and from work is a bus. Suppose that in a large city P(A) = 0.44 and P(B) = 0.32. A.Are events A and B mutually exclusive? B. What is the probability that a person uses an automobile or a bus in going to and from work?

A. Yes B. .76 C. .68

What do you do when you see OR

Add

Sum of Probabilities must.......

Add up to 1

Three Methods To Assign Probabilities

Classical Method: If an experiment has n possible outcomes, the classical method would assign a probailit of 1/n to each outcome Relative frequency method: probabilities can be based on experimentation or historical data Subjective Method: Probabilities can be based on judgment

Event

Collection of Sample Points

4 basic probability relationships

Complement of an event Addition Law Conditional probability Multiplication law

Independent Events + independent if

- If the probability of event A is not changed by the existence of event B, we would say that events A and B are independent. - Independent If P(AIB)=P(A) or P(BIA)=P(B)

Equation for the Complement of an Event

- The complement of event A is defined to be the event consisting of all sample points that are not in A. - P(A) = 1- P(A^c)

Intersection of Two events

- The intersection of events A and B is the set of all sample points that are in both A and B. - M(upside down U)C = Markley Oil Profitable and Collins Mining Profitable

Conditional Probability + equation

- The probability of an event given that another event has occurred is called a conditional probability - The conditional probability of A given B is denoted by P(A|B). -P(AIB)=P(A(int)B/P(B)

Union of Two events What Equation

- The union of events A and B is the event containing all sample points that are in A or B or both. - The union of events A and B is denoted by AuB - Ex. Event M = Markley Oil Profitable Event C =Collins Mining Profitable MUC = Markley Oil Profitable or Collins Mining Profitable (or both) - union of two events

Additon law + Equation

-The addition law provides a way to compute the probability of event A, or B, or both A and B occurring. - P(AUB)=P(A)+P(B)-P(A (intersect) B) - For Union of two events

Multiplication Law + Equation

-The multiplication law also can be used as a test to see if two events are independent. P(A (int) 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. - Two events are mutually exclusive if, when one event occurs, the other cannot occur.

A pharmaceutical company conducted a study to evaluate the effect of an allergy relief medicine; 250 patients with symptoms that included itchy eyes and a skin rash received the new drug. The results of the study are as follows: 85 of the patients treated experienced eye relief, 140 had their skin rash clear up, and 50 experienced relief of both itchy eyes and the skin rash. What is the probability that a patient who takes the drug will experience relief of at least one of the two symptoms?

.7

Multiplication Law +equation

P(A (int) B)= P(B)*P(AIB)

If events are mutually exclusive the addition law is what?

P(AUB)= P(A)+P(B)

Probability of an event =

Sum of probabilities of the sample points in the event

Probability

a Numerical measure of the liklihood that an event will occor -Prob Values are always assigned on a scale of 0-1 - a Prob near 0 indicates an event is not likely to occur - A prob near one indicates an event is almost certain to occur

Sample space for an experiment Example: Experiment = toss a coin

the set of all experimental outcomes or sample points Sample space: - Head, Tail

4 rules for Mutual Exclusiveness and independence

Do not confuse the notion of mutually exclusive events with that of independent events. Two events with nonzero probabilities cannot be both mutually exclusive and independent. If one mutually exclusive event is known to occur, the other cannot occur.; thus, the probability of the other event occurring is reduced to zero (and they are therefore dependent). Two events that are not mutually exclusive, might or might not be independent.

How do you set up a joint Probability table

Draw it out