Unit 3:Probability

Definitions:

Randomness: a phenomenon is random if individual outcomes are uncertain, but there is nonetheless a regular distribution of outcomes in a large number of repetitions.

-The probability of any outcome of a random phenomenon can be defined as the proportion of times the outcome would occur in a very long series of repetitions

~Actual probability: experimental probability is found by collecting date

~Theoretical probability: what we expect to happen

Law of large numbers:as the number of trials increases in a probability experiment the actual probability will converge on the theoretical probability

Probability: number of outcomes that are favorable

-total number of outcomes

Sample space: set of all possible outcomes

Notation:

-A= event A is that a student is a senior

-P(A)= Probability of event A

-P(senior)= Probability of the event that a student is a senior

Complements:

-A= seniors -A^c=Not seniors

-P(A^c)= Probability of the complement of event A

-P(A^c)=1-P(A)

Facts about probability: 0<P<1

-P(A)= 0 event A is impossible

-P(B)= 1 event B is certain to happen

Venn Diagrams:

Making Diagrams:

8/18

3/18

5/18

2/18

Our class has students

11 are female(F)

13 are seniors (A)

8 are female and seniors (F and A)

Tree Diagram:

Table:

Tree Diagram: best for information that is sequential

Venn Diagram: best for joint information occurring concurrently

Table: universal to use but not always the most efficient method

Or Statements:

Definitions:

Conditional Statement:

U=Union (or)

Upside down U= intersection (and)

P(A U B)=P(A)+P(B)-P(A upsidedownU B)

P(A or B)=P(A)+P(B)-P(A and B)

Mutually Exclusive/Disjoint Events:

-Events are mutually exclusive if they cannot occur at the same time

-disjoint is another word to describe mutually exclusive. If 2 events A and B are disjoint then P(a upsidedownU B)=O

Independence: the probability of an event is not changed by a previous event

-Replacement --> independent

-No replacement --> dependent

Conditional events: probability of A given B

-P(A|B)

~"|" = given

~ whatever is after the vertical line is the part that's given

And Statements (dependent events):

- upsidedownU=intersection

-Intersection means and

~P(A upsidedownU B)=P(A)*P(B|A)

Thing that is given is the denomenator

P(A upsidedownU B)=P(A)*P(B) if and only if A and B are independent events

Proving independent events:

Show one of these three equations are true

P(A upsidedownU B)=P(A)*P(B)

P(B|A)=P(B)

P(A|B)=P(A)