1. Section 6.2 Probability Models
AP Statistics
Section 6.2 Probability Models

2. 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:
Empirical: probability based on observation.
Ex. Hershey Kisses:

3. 2. Theoretical: probability based on a mathematical model. Ex
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.

4. Methods for Finding the Total Number of Outcomes:
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.
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.

5. 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.

6. 𝑃 𝐴 = 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 𝐴 𝑜𝑐𝑐𝑢𝑟𝑠 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 0≤𝑃 𝐴 ≤1
Rules for Probability
Let A = any event; Let P(A) be read as “the probability of event A”
𝑃 𝐴 = 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 𝐴 𝑜𝑐𝑐𝑢𝑟𝑠 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
0≤𝑃 𝐴 ≤1
If P(A) = 0 then A can never occur.
If P(A) = 1 then A always occurs.
𝑃 𝐴 =1; the sum of all the probabilities of the outcomes in S equals 1.
Complement Rule: 𝐴 𝑐 or 𝐴 ′ is read as “the complement of A”
𝑃 𝐴 𝑐 is read as “the probability that A does NOT occur”
𝑃 𝐴 𝑐 =1−𝑃(𝐴) or 𝑃 𝐴 𝑐 +𝑃 𝐴 =1
Key words: not, at least, at most

7. Ex. 1 Roll one die, find 𝑃 6 𝑐 Ex
Ex. 1 Roll one die, find 𝑃 6 𝑐 Ex. 2 Flip 5 coins, find P(at least 1 tail)

8. The General Addition Rule: (use when selecting one item)
𝑃 𝐴 𝑜𝑟𝐵 =𝑃 𝐴 +𝑃 𝐵 −𝑃(𝐴 𝑎𝑛𝑑 𝐵)
𝑃 𝐴∪𝐵 =𝑃 𝐴 +𝑃 𝐵 −𝑃(𝐴 ∩𝐵)
Ex. Roll one die, find 𝑃(<3 𝑜𝑟 𝐸𝑣𝑒𝑛)
Ex. Roll one die, find 𝑃(<3 𝑜𝑟>4)
Events A and B are disjoint if A and B have no elements in common. (mutually exclusive)
𝑃 𝐴 𝑎𝑛𝑑 𝐵 =0
𝐴∩𝐵=∅

9. Ex. Choose one card from a standard deck of cards
Ex. Choose one card from a standard deck of cards. Find 𝑃 𝑅𝑒𝑑 𝑜𝑟 𝐾𝑖𝑛𝑔 𝑃 𝐷𝑖𝑎𝑚𝑜𝑛𝑑 𝑜𝑟 8 𝑃(𝐹𝑎𝑐𝑒 𝑐𝑎𝑟𝑑 𝑜𝑟 𝑏𝑙𝑎𝑐𝑘) 𝑃 𝑘𝑖𝑛𝑔 𝑎𝑛𝑑 𝑞𝑢𝑒𝑒𝑛 𝑃(𝐻𝑒𝑎𝑟𝑡 𝑜𝑟 𝑆𝑝𝑎𝑑𝑒) 𝑃 𝑃𝑖𝑐𝑡𝑢𝑟𝑒 𝑐𝑎𝑟𝑑 𝑜𝑟 10

10. Equally Likely Outcomes: If sample space S has k equally likely outcomes and event A consists of one of these outcomes, then 𝑃 𝐴 = 1 𝑘 Ex.
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 𝑃 𝐴 𝑎𝑛𝑑 𝐵 =𝑃 𝐴 ∙𝑃 𝐵
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.

11. Ex. TEST FOR INDEPENDENCE
Ex. TEST FOR INDEPENDENCE. Flip 2 coins, let A = heads on 1st and B = heads on 2nd. Are A and B independent? Find 𝑃(𝐴 𝑎𝑛𝑑 𝐵) Find 𝑃(𝐴)∙𝑃(𝐵) Any events that involve “replacement” are independent and events that involve “without replacement” are dependent.

12. Ex. Choose 2 cards with replacement from a standard deck
Ex. Choose 2 cards with replacement from a standard deck. Find 𝑃(𝐴𝑐𝑒 𝑎𝑛𝑑 𝐾𝑖𝑛𝑔) 𝑃(10 𝑎𝑛𝑑 𝐹𝑎𝑐𝑒 𝐶𝑎𝑟𝑑) Repeat without replacement:

13. IF EVENTS ARE DISJOINT, THEN THEY CAN NOT BE INDEPENDENT. Ex
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?

14. Independence vs. Disjoint
Case 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)

15. Case 2) A and B are NOT disjoint and dependent
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)

16. Case 3) A and B are disjoint and dependent. Given P(A) = 0
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

17. 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
0.25 7
0.05