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13.4 Determining Probabilities

Essential Question: What are the two types of probability?. 13.4 Determining Probabilities. 13.4 Determining Probabilities. Exact probability of a real event can never be known Probabilities are estimated in two ways: experimentally and theoretically

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13.4 Determining Probabilities

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  1. Essential Question: What are the two types of probability? 13.4 Determining Probabilities

  2. 13.4 Determining Probabilities • 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

  3. 13.4 Determining Probabilities • 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

  4. 13.4 Determining Probabilities • 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)

  5. 13.4 Determining Probabilities • Theoretical Estimates of Probability • Example 3: Rolling a number cubeAn experiment consists of rolling a number cube. Assume all outcomes are equally likely. • Write the probability distribution for the experiment. • Find the probability of the event that an even number is rolled. P(2, 4, or 6) = 1/6 + 1/6 + 1/6 = 3/6 = 1/2

  6. 13.4 Determining Probabilities • 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

  7. 13.4 Determining Probabilities • Homework: Page 882, 1-17 (ALL) • We’ll run the experiment for numbers 8-11 as a class

  8. 13.4 Determining Probabilities • 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

  9. 13.4 Determining Probabilities • Example: Choosing 3 letters of the alphabet • Two important factors in determining probability: • Are selections chosen with replacement? • Is it possible to come up with the outcome ‘AAA’ or not? • Is order important? • Is there a difference between ‘EAT’ and ‘ATE’?

  10. 13.4 Determining Probabilities • Permutation: • Combination:

  11. 13.4 Determining Probabilities • Choosing 3 letters of the alphabet • With replacement, order important • 263 = 17,576 • Without replacement, order important • Without replacement, order matters

  12. 13.4 Determining Probabilities • 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: • There’s only one possibility where they’re sent correctly • The number of possible outcomes (because order matters) is 4P4 = 24 • The probability is 1/24 ≈ 0.04

  13. 13.4 Determining Probabilities • Example 8: Pick-6 Lottery • 54 numbered balls are used; 6 are randomly chosen. • To win anything, at least 3 numbers must match. • What’s the probability of matching all 6? 5 of 6? 4 of 6? 3 of 6? • Answer: • Order doesn’t matter, and numbers aren’t replaced • 54C6 = 25,827,165

  14. 13.4 Determining Probabilities • Example 8: Pick-6 Lottery (Answer) • Answer: • Order doesn’t matter, and numbers aren’t replaced • Total combinations: 54C6 = 25,827,165 • P(jackpot) = 1/25,827,165 • P(5 correct) = (6C5 ∙ 48C1)/25,827,165=288/25,827,165 • P(4 correct) = (6C4 ∙ 48C2)/25,827,165=16,920/25,827,165 • P(3 correct) = (6C3 ∙ 48C3)/25,827,165=345,920/25,827,165 • P(win anything) = 363,129/25,827,165 ≈ 0.014

  15. 13.4 Determining Probabilities • Homework: Page 882, 18-29 (ALL)

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