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6.1-6.2 Probability Models
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Understand the term “random” Implement different probability models Use the rules of probability in calculations
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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
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- 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.
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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.
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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 xHT P(x)½½
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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
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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….
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Resembles the branches of a tree. *allows us to not overlook things
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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
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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 01234 TTTTHTTTHHTTTHHHHHHH THTTHTHTHTHH TTHTHTTHHHTH TTTHTHHTHHHT TTHH THTH
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X01234 P(x)1/161/43/81/41/16
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Ex: Generate a random decimal number. What is the sample space? S={all numbers between 0 and 1}
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a) S= {G,F} b) S={length of time after treatment} c) S={A,B,C,D,F}
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With replacement- same probability and the events remain independent Ex: Without replacement- changes the probability of an event occurring Ex:
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#1) 0 ≤ P(A) ≤ 1 #2) P(S) = 1
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#3- #4- Disjoint- A and B have no outcomes in common (mutually exclusive) P(A or B)= P(A) + P(B)
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Union: “or” P(A or B) = P(A U B) Intersect: “and” P(A and B) = P(A ∩ B) Empty event: (no possible outcomes) S={ } or ∅
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P(A)= 0.34 P(B)=0.25 P(A ∩ B)=0.12
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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 Marital Status Never Married WidowedDivorced Probability0.3530.5740.0020.071
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A= {first digit is 1} P(A)=.30 B= {first digit is 6 or greater} P(B)=.222 C={first digit is greater than 6} P(C)=.155 First Digit 123456789 Prob..301.176.125.097.079.067.058.051.046
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D={first digit is not 1} P(D)= 1- 0.301= 0.699 E={1st number is 1, or 6 or greater} P(E)=0.522 F={ODD} P(F)=0.609 G={odd or 6 or greater} P(G)=0.727
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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
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Try 6.18 with your partners A) 0.04 B) 0.69 Try 6.19 A) 0.1 B) 0.3 C) regular: 0.5 peanut: 0.4
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Rule 5: P(A and B)= P(A) P(B) (only for independent events!)
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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
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