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Understanding Probability Models in AP Statistics: Randomness and Events

This resource covers the fundamentals of probability models in AP Statistics, focusing on the concept of randomness and the implementation of various probability models. Students will learn how outcomes are unpredictable in the short run but tend to exhibit a predictable pattern over numerous repetitions. Key topics include the definition of a sample space, events, and the rules of probability, such as the multiplication rule for independent events. Through examples, simulations, and visual tools like tree diagrams and Venn diagrams, learners can enhance their grasp of probability.

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Understanding Probability Models in AP Statistics: Randomness and Events

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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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