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Section 8.2 Geometric Distributions

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

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Section 8.2 Geometric Distributions

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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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