Chapter 6: Probability

Adam TurryJordan Furr

Ansley OrgeronMackenzie Kruse

What is a simulation?

• A simulation is the imitation of chance behavior, based on a model that accurately reflects the phenomenon under consideration.

• The calculator is a great way to carry out a simulation by using random integer

Steps of Simulations

• State the problem• State the assumptions• Assign digits to represent outcomes• Simulate multiple repetions• Say your conclusion

Probability Models

• A random phenomenon has outcomes that we cannot predict but have a regular distribution with many repetitions.

• The probability of an event is the proportion of times the event occurs in many repeated trials of a random phenomenon.

• A probability model for a random phenomenon consists of a sample space S and an assignment of probabilities.

• The sample space S is the set of all possible outcomes of the random phenomenon.

• Events are sets of outcomes.• A number P(A) is assigned to an event A as its probability.

Vocabulary

• A tree diagram is a simple way to look at the possible outcomes of a probability sample. Each of the outcomes is a branch on the “tree.”

• The multiplication principle says is you can do one task in n1 number of ways and another task in n2 number of ways, then both tasks can be done in n1 x n2 number of ways.

More Vocabulary• Sampling with replacement takes place when you select an object and

replace it before the next selection.– Replacing a marble into the bag before picking another

• Sampling without replacement, the probabilities change for each new selection.– Taking a marble out of the bag and leaving it out before selecting another.

• The complement Ac of an event A consists of the outcomes that are not in A.

• Events are disjoint (mutually exclusive) if they have no outcomes in common.

• Events are independent if knowing that one event occurs does not change the probability we would assign the other event.

• A Venn diagram shows events as disjoint of intersecting regions.

Rules of Probability

1. Legitimate Values- 0≤P(A) ≤1 for any event A2. Total Probability- P(S) =1 for the sample space S.3. Addition rule: if events a and B are disjoint, then P(A

or B) = P(A U B) = P(A) + P(B).4. Complement rule: For any event A, P(Ac) = 1- P(A).5. Multiplication rule : if events A and B are

independent, then P(A and B) = P (A∩B) = P(A)P(B).6. General Addition Rule- (A U B) = P(A)+P(B)- (A∩B)

General Probability Rules• Intersection- (A∩B)

contains all the outcomes that are in set A, and B

• Union-(A U B) contains all outcome of A or B, or in both A and B.

• Conditional Probability- the event of P(B) occurring given P(A) has occurred. Formula= P(A∩B)/P(A)

Venn Diagram

• Venn diagrams are word problems that generally help you find probabilities of the union and/or intersection of two events.

• Example:

Helpful Hints

• Disjoint(mutually exclusive)- if A and B are disjoint, the P(A∩B)=0– The general addition rule for

unions is then the special addition rule P(A U B)= P(A) + P(B)

• A and B are independent when P(B│A)=P(B)– Multiplication rule for

intersection then becomes P(A∩B)=P(A) P(B)