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Essential Question: What are the two types of probability?

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Exact probability of a real event can never be known Probabilities are estimated in two ways: experimentally and theoretically Experimental probability is done by running experiments and calculating the results Theoretical probability is done by making assumptions on the results

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Example 1: Experimental Estimate of Probability Throw a dart at a dartboard Red: 43, Yellow: 86, Blue: 71 Write a probability distribution for the experiment Outcomeredyellowblue Probability

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Probability Simulation In order for experimental probability to be useful, a large number of simulations need to be run Computer simulations, done via random number generators, prove useful Example 2: Probability Solution Flip a coin 3 times, and count the number of heads (In-class simulation)

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Theoretical Estimates of Probability Example 3: Rolling a number cube An experiment consists of rolling a number cube. Assume all outcomes are equally likely. a)Write the probability distribution for the experiment. b)Find the probability of the event that an even number is rolled. Outcome123456 Probability1616 1616 1616 1616 1616 1616 P(2, 4, or 6) = 1/6 + 1/6 + 1/6 = 3/6 = 1/2

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Example 4: Theoretical Probability Find the theoretical probability for the experiment you ran in Example 2 (flipping 3 coins) 0 heads: only 1 possible outcome (TTT) 0.5 x 0.5 x 0.5 = 0.125 1 head: 3 possible outcomes (HTT, THT, TTH) 3 x 0.125 = 0.375 2 heads: 3 possible outcomes (HHT, HTH, THH) 3 x 0.125 = 0.375 3 heads: only 1 possible outcome (HHH) 0.5 x 0.5 x 0.5 = 0.125

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Homework: Page 882, 1-17 (ALL) Well run the experiment for numbers 8-11 as a class

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Counting Techniques If a set of experiments have multiple potential outcomes each, the total number of outcomes is simply the product of the individual outcomes Example 5 A catalog offers chairs in a choice of 2 heights. There are 10 colors available for the finish, and 12 choices of fabric for the seats. The chair back has 4 different possible designs. How many different chairs can be ordered? Answer: 2 10 12 4 = 960 possible outcomes

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Example: Choosing 3 letters of the alphabet Two important factors in determining probability: 1. Are selections chosen with replacement? Is it possible to come up with the outcome AAA or not? 2. Is order important? Is there a difference between EAT and ATE?

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With replacementWithout replacement Order mattersxnxn Permutation (nPr) Order unimportant(wont be discussed)Combination (nCr) Permutation: Combination:

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Choosing 3 letters of the alphabet 1. With replacement, order important 26 3 = 17,576 2. Without replacement, order important 3. Without replacement, order matters

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Example 7: Matching Problem Suppose you have four personalized letters and four addressed envelopes. If the letters are randomly placed in the envelopes, what is the probability that all four letters will go to the correct address? Answer: Theres only one possibility where theyre sent correctly The number of possible outcomes (because order matters) is 4 P 4 = 24 The probability is 1/24 0.04

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Example 8: Pick-6 Lottery 54 numbered balls are used; 6 are randomly chosen. To win anything, at least 3 numbers must match. Whats the probability of matching all 6? 5 of 6? 4 of 6? 3 of 6? Answer: Order doesnt matter, and numbers arent replaced 54 C 6 = 25,827,165

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Example 8: Pick-6 Lottery (Answer) Answer: Order doesnt matter, and numbers arent replaced Total combinations: 54 C 6 = 25,827,165 P(jackpot) = 1/25,827,165 P(5 correct) = ( 6 C 5 48 C 1 )/25,827,165=288/25,827,165 P(4 correct) = ( 6 C 4 48 C 2 )/25,827,165=16,920/25,827,165 P(3 correct) = ( 6 C 3 48 C 3 )/25,827,165=345,920/25,827,165 P(win anything) = 363,129/25,827,165 0.014

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Homework: Page 882, 18-29 (ALL)

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