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Geometric Distributions in AP Statistics

Learn how to calculate probabilities and expectations using geometric distributions in AP Statistics. Explore examples and formulas for practical application.

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Geometric Distributions in AP Statistics

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  1. Section 8.2Geometric Distributions AP Statistics December 12, 2012

  2. The Geometric Setting • Each observation falls into one of just two categories, which for convenience we call “success” or “failure” • You keep trying until get a success • The observations are all independent. • The probability of success, call it p, is the same for each observation. AP Statistics, Section 8.2.2

  3. Calculating Probabilities • The probability of rolling a 6=1/6 • The probability of rolling the first 6 on the first roll: • P(X=1)=1/6. • geometpdf(1/6,1) • The probability of rolling the first 6 after the first roll: • P(X>1)=1-1/6 = 5/6 • 1-geometpdf(1/6,1) AP Statistics, Section 8.2.2

  4. Calculating Probabilities • The probability of rolling a 6=1/6 • The probability of rolling the first 6 on the second roll: • P(X=2)=(1/6)*(5/6). • geometpdf(1/6,2) • The probability of rolling the first 6 on the second roll or before: • P(X<2)=(1/6) +(1/6)*(5/6) • geometcdf(1/6,2) AP Statistics, Section 8.2.2

  5. Calculating Probabilities • The probability of rolling a 6=1/6 • The probability of rolling the first 6 on the second roll: • P(X=2)=(1/6)*(5/6). • geometpdf(1/6,2) • The probability of rolling the first 6 after the second roll: • P(X>2)=1-((1/6) +(5/6)*(1/6)) • 1-geometcdf(1/6,2) AP Statistics, Section 8.2.2

  6. You can use these formulas AP Statistics, Section 8.2.2

  7. Formulas for Geometric Distribution AP Statistics, Section 8.2.2

  8. Example • In New York City at rush hour, the chance that a taxicab passes someone and is available is 15%. • a) How many cabs can you expect to pass you for you to find one that is free • 6.67 so 7 cabs • b) what is the probability that more than 10 cabs pass you before you find one that is free. • 19.68%

  9. Histograms of Geometric distributions For illustration, we will use the roll of a die with n = 6 likely outcomes, and probability p = 1/6 of rolling a 3 (our success). The random variable X is the number of rolls until a 3 is observed. Enter numbers 1-10 in L1 L2 is geometpdf(1/6, L1) ENTER. L3 you can make geometcdf(1/6, L1)ENTER Do the histograms as before

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