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AP Statistics

AP Statistics. 6.1-6.2 Probability Models. Learning Objective:. Understand the term “random” Implement different probability models Use the rules of probability in calculations. Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run

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AP Statistics

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  1. AP Statistics 6.1-6.2 Probability Models

  2. Learning Objective: • Understand the term “random” • Implement different probability models • Use the rules of probability in calculations

  3. Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run • What does that mean to you? the more repetition, the closer it gets to the true proportion

  4. Random • - if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions. • 1- you must have a long series of independent trials • 2- probabilities imitate random behavior • 3- we use a RDT or calculator to simulate behavior.

  5. Probability • The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions. That is, the probability is long-term relative frequency.

  6. 6.2 Probability Models • What is a mathematical description or model for randomness of tossing a coin? • This description has two parts. • 1- A list of all possible outcomes • 2- A probability for each outcome

  7. Probability Models • Sample space S- a list of all possible outcomes. • Ex: S= {H,T} S={0,1,2,3,4,5,6,7,8,9} • Event- an outcome or set of outcomes (a subset of the sample space) • Ex: roll a 2 when tossing a number cube

  8. Example: • If we have two dice, how many combinations can you have? 6 * 6 = 36 • If you roll a five, what could the dice read? (1,4) (4,1) (2,3) (3,2) • How can we show possible outcomes? list, tree diagram, table, etc….

  9. Tree Diagram- • Resembles the branches of a tree. *allows us to not overlook things

  10. Multiplication Principle- • If you can do one task in a number of ways and a second task in b number of ways, then both tasks can be done in a x b number of ways. • Ex: How many outcomes are in a sample space if you toss a coin and roll a dice? 2 * 6 = 12

  11. Ex: You flip four coins, what is your sample space of getting a head and what are the possible outcomes? S= {0,1,2,3,4} Possible outcomes: 2 * 2 * 2 * 2 = 16

  12. What is the probability Distribution?

  13. Example • Ex: Generate a random decimal number. What is the sample space? • S={all numbers between 0 and 1}

  14. Pg. 322: 6.9 a) S= {G,F} b) S={length of time after treatment} c) S={A,B,C,D,F}

  15. With replacement- same probability and the events remain independent • Ex: • Without replacement- changes the probability of an event occurring • Ex:

  16. Probability Rules • #1) 0 ≤ P(A) ≤ 1 • #2) P(S) = 1

  17. Probability Rules • #3- • #4- Disjoint- A and B have no outcomes in common (mutually exclusive) P(A or B)= P(A) + P(B)

  18. Union: “or” P(A or B) = P(A U B) • Intersect: “and” P(A and B) = P(A П B) • Empty event: S={ } or ∅

  19. Venn Diagram:

  20. Display the probabilities by using a Venn Diagram. • P(A)= 0.34 • P(B)=0.25 • P(A П B)=0.12

  21. Marital Status • What is the sum of these probabilities? 1 • P(not married)= 1- P(M)= 1 – 0.574 = 0.426 • P(never married or divorced)= 0.353 + 0.071 = 0.424

  22. Probabilities in a finite sample space: • Assign a probability to each individual outcome. These probabilities must be numbers between 0 and 1 and must have sum 1. • The probability of any event is the sum of the probabilities of the outcomes making up the event.

  23. Benford’s Law • A= {first digit is 1} P(A)= • B= {first digit is 6 or greater} P(B)= • C={first digit is greater than 6} P(C)= • D={first digit is not 1} P(D)= • E={1st number is 1, or 6 or greater} P(E)= • F={ODD} P(F)=

  24. Equally likely outcomes • If a random phenomenon has k possible outcomes, all equally likely, then each individual outcome has probability 1/k.The probability of any event A is: • P(A)= count of outcomes in A count of outcomes in S

  25. Pg. 330: 6.18, 6.19

  26. The Multiplication Rule for Independent Events • Rule 5: P(A and B)= P(A) P(B) (only for independent events!)

  27. Pg. 335 6.24: One Big: 0.6 3 small: (0.8)³=0.512 6.25: (1-0.05)^12=0.5404 6.26: the events aren’t independent 6.27:

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