AP STATISTICSSection 6.2 Probability Models

Objective: To be able to understand and apply the rules for probability.

Random: refers to the type of order that reveals itself after a large number of trials.

Probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions.

Types of Probability:

1. Empirical: probability based on observation.

Ex. Hershey Kisses:

2. Theoretical: probability based on a mathematical model.

Ex. Calculate the probability of flipping 3 coins and getting all head.

Sample Space: set of all possible outcomes of a random phenomenon.

Outcome: one result of a situation involving uncertainty.

Event: any single outcome or collection of outcomes from the sample space.

Methods for Finding the Total Number of Outcomes:

1. Tree Diagrams: useful method to list all outcomes in the sample space. Best with a small number of outcomes.

Ex. Draw a tree diagram and list the sample space for the event where one coin is flipped and one die is rolled.

2. Multiplication Principle: If event 1 occurs M ways and event 2 occurs N ways then events 1 and 2 occur in succession M*N ways.

Ex. Use the multiplication principle to determine the number of outcomes in the sample space for when 5 dice are rolled.

Sampling with replacement: when multiple items are being selected, the previous item is replaced prior to the next selection.

Sampling without replacement: then the item is NOT replaced prior to the next selection.

Rules for ProbabilityLet A = any event; Let P(A) be read as “the probability of event A”

1.

2.

3. If P(A) = 0 then A can never occur.

4. If P(A) = 1 then A always occurs.

5. ; the sum of all the outcomes in S equals 1.

6. Complement Rule: or is read as “the complement of A”

is read as “the probability that A does NOT occur”

or

Key words: not, at least, at most

Ex. 1 Roll one die, find

Ex. 2 Flip 5 coins, find P(at least 1 tail)

7. The General Addition Rule: (use when selecting one item)

Ex. Roll one die, find

Ex. Roll one die, find

Events A and B are disjoint if A and B have no elements in common. (mutually exclusive)

Ex. Choose one card from a standard deck of cards. Find

8. Equally Likely Outcomes: If sample space S has k equally likely outcomes and event A consists of one of these outcomes, then Ex.

9. The Multiplication Rule: (use when more than one item is being selected)

If events A and B are independent and A and B occur in succession, the

Events A and B are said to be independent if the occurrence of the first event does not change the probability of the second event occurring.

Ex. TEST FOR INDEPENDENCE. Flip 2 coins, let A = heads on 1st and B = heads on 2nd. Are A and B independent?

Find Find

Any events that involve “replacement” are independent and events that involve “without replacement” are dependent.

Ex. Choose 2 cards with replacement from a standard deck. Find

Repeat without replacement:

IF EVENTS ARE DISJOINT, THEN THEY CAN NOT BE INDEPENDENT!!!!!

Ex. Let A = earn an A in Statistics; P(A) = 0.30

Let B = earn a B in Statistics; P(B) = 0.40

Are events A and B disjoint?

Are events A and B independent?

Independence vs. DisjointCase 1) A and B are NOT disjoint and independent.

Suppose a family plans on having 2 children and the P(boy) = 0.5

Let A = first child is a boy. Let B = second child is a boy

Are A and B disjoint?

Are A and B independent? (check mathematically)

Case 2) A and B are NOT disjoint and dependent. (Use a Venn Diagram for Ex)

Are A and B disjoint?

Are A and B independent? (check mathematically)

Case 3) A and B are disjoint and dependent.

Given P(A) = 0.2 , P(B) = 0.3 and P(A and B) = 0

Are A and B independent? (check mathematically)

(Also refer to example for grade in class)

Case 4) A and B are disjoint and independent.

IMPOSSIBLE

Ex. Given the following table of information regarding meal plan and number of days at a university:

A student is chosen at random from this university, find

P(plan A) P(5 days)

P(plan B and 2 days) P(plan B or 2 days)

Are days and meal plan independent? (verify mathematically)

Day/Meal Plan A Plan B Total:

2 0.15 0.20

5 0.20 0.25

7 0.05 0.15

Total: