Basic Probability Sets, Subsets Sample Space Event, E Probability of an Event, P(E) How Probabilities are assigned Properties of Probabilities

Sets A set is simply any collection of objects A set may be finite or infinite A set with nothing in it is called the empty set (null or void set) and is denoted, { } or ø Two sets are equal if they have exactly the same elements

Subsets If A={1,2,3} is a set then subsets of A include the sets:{ },{1},{2},{3},{1,2}, {1,3},{2,3},{1,2,3} If B is one of the subsets of A then we can say that B  A

Sample Space The set, S, of all distinct possible outcomes of an experiment is called a sample space Suppose we are rolling a die, what is the sample space, S? Suppose we toss a coin twice recording the outcome each time, what is the sample space, S?

Events (E) An event is any collection of outcomes of a probability experiment Suppose we are flipping a coin-what are the events that may occur? Suppose we are rolling a die, what are the events that may occur? What if we flip the coin twice, what are the events that may occur?

Event, con’t Since an event is a collection of outcomes, we can say that E  S What does E mean? What does S stand for? What does E  S mean? An event, E is a subset of a sample space, S.

Exercise Determine the sample space for the experiment: flipping a coin three times Write three events that correspond to this experiment What is P(at least 3 heads)? What is P(at least 3 tails)? What is P(no heads)? What is P(no more than 1 head)?

Another Exercise Suppose as part of a survey on popular music two students are asked whether they like a certain CD, dislike it, or don’t care. What is the sample space? What are the chances that the first student likes the CD? What are the chances that the second student likes the CD?

Another Exercise (cont) How about each of the students either liking or disliking the CD? How about one student liking the CD while the other doesn’t care? What about the second student liking the CD AND one of the students likes the CD while the other doesn’t care?

Probability of an Event Given an event, we would assign it a number, P(E) called the probability of E This number indicates the likelihood that the event will occur. We can find this number by setting up a ratio:

How Probabilities are Assigned Initial Probabilities are usually assigned either: Empirically-when an experiment is repeated a large number of times and you observe the fraction of times E occurs By Authority By Common Agreement

Properties of Probabilities Probabilities must satisfy the following properties: i. For any event, E, 0  P(E)  1 ii. If E is certain to happen then P(E)=1 iii. If E and F are events where E and F cannot happen at the same time, then P(E or F)=P(E)+P(F)

Probability, con’t A set of possible outcomes of an experiment is a sample space, S. P(S)=?

Venn Diagrams The Venn Diagram is made up of two or more overlapping circles or sets. It is often used in mathematics to show relationships between sets.

Venn Diagrams Here is the Venn Diagram associated with the set A.

Complements A complement of A is everything that is in the universal set, U, but not in the set A. The complement is the event that A does not happen. The complement is denoted, A c. Here is the complement of set A.

Unions of sets The union of sets A and B is the set of all items that are either in A or B. We express union, A  B In math, the word “or” also includes members of both A and B.

Intersection of Sets The intersection of sets A and B is the set of all items that are in both A and B. We express intersection, A  B.

Probabilities For any events E and F:

Mutually Exclusive Two events are mutually exclusive if A  B= . If A and B are mutually exclusive, then AB

Mutually Exclusive If no two events E 1,E E n can happen at the same time, then

Review Union of two sets, A  B, is all items that are in set A OR set B Intersection of two sets, A  B, is all items that are in both A AND B A complement of event A, A c, is everything that is in the universal set but not in the set A Two events are mutually exclusive if A  B= 

Review, con’t For any events, E and F: P(E  F)=P(E)+P(F)-P(E  F) We can use this equation to solve for P(E  F), P(E), P(F), or P(E  F) However if E and F are mutually exclusive then: P(E  F)=P(E)+P(F) P(E c )=1-P(E)

De Morgan’s Laws If A and B are any two sets:

Answer the following questions: What is: (A  A c ) = ? (A  A c ) = ? If E and F are mutually exclusive, what is (E  F) = ?