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AP STATISTICS Section 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 Probability Let A = any event; 1. π π΄ = Let P(A) be read as βthe probability of event Aβ π‘βπ ππ’ππππ ππ π€ππ¦π π΄ ππππ’ππ π‘ππ‘ππ ππ’ππππ ππ ππ’π‘πππππ 0β€π π΄ β€1 3. If P(A) = 0 then A can never occur. 4. If P(A) = 1 then A always occurs. 5. π π΄ = 1; the sum of all the outcomes in S equals 1. 6. Complement Rule: π΄π or π΄β² is read as βthe complement of Aβ π π΄π is read as βthe probability that A does NOT occurβ π π΄π = 1 β π(π΄) or π π΄π + π π΄ = 1 Key words: not, at least, at most 2. Ex. 1 Roll one die, find π 6π Ex. 2 Flip 5 coins, find P(at least 1 tail) 7. The General Addition Rule: (use when selecting one item) π π΄ πππ΅ = π π΄ + π π΅ β π(π΄ πππ π΅) π π΄ βͺ π΅ = π π΄ + π π΅ β π(π΄ β© π΅) Ex. Roll one die, find π(< 3 ππ πΈπ£ππ) Ex. Roll one die, find π(< 3 ππ > 4) Events A and B are disjoint if A and B have no elements in common. (mutually exclusive) π π΄ πππ π΅ = 0 π΄β©π΅ =β Ex. Choose one card from a standard deck of cards. Find π π ππ ππ πΎπππ π π·ππππππ ππ 8 π(πΉπππ ππππ ππ πππππ) π ππππ πππ ππ’πππ π(π»ππππ‘ ππ πππππ) π ππππ‘π’ππ ππππ ππ 10 8. Equally Likely Outcomes: If sample space S has k equally likely outcomes and event A consists of one of 1 these outcomes, then π π΄ = Ex. π 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 π π΄ πππ π΅ = π π΄ β π π΅ 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 π(π΄ πππ π΅) Find π(π΄) β π(π΅) Any events that involve βreplacementβ are independent and events that involve βwithout replacementβ are dependent. Ex. Choose 2 cards with replacement from a standard deck. Find π(π΄ππ πππ πΎπππ) π(10 πππ πΉπππ πΆπππ) Repeat without replacement: π(π΄ππ πππ πΎπππ) π(10 πππ πΉπππ πΆπππ) 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. 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) 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: Day/Meal Plan A Plan B 2 0.15 0.20 5 0.20 0.25 7 0.05 0.15 Total: Total: 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)