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Chapter 17 – Probability Models

Chapter 17 – Probability Models. Multiple Choice Test. Suppose we have a multiple choice test where each question has 5 choices. If we had a 4 question test, find the probability that : We get all 4 wrong The first one we get correct is the 4 th. Multiple Choice Test continued.

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Chapter 17 – Probability Models

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  1. Chapter 17 – Probability Models

  2. Multiple Choice Test • Suppose we have a multiple choice test where each question has 5 choices. • If we had a 4 question test, find the probability that: • We get all 4 wrong • The first one we get correct is the 4th

  3. Multiple Choice Test continued Suppose we have a multiple choice test with 4 questions, each with 5 choices. Find the probability we get: Only the first one correct Only the second correct Only the third correct Only the fourthcorrect Exactly one of the 4 correct

  4. Bernoulli Trials • 2 possible outcomes (success and failure) • Probability of success, p, is constant • Trials are independent • Often we don’t technically have independence, but if the sample is less than 10% of population, it’s ok to use Bernoulli trials

  5. Binomial Model Binomial model used to count the number of successes in a fixed number of Bernoulli trials Binom(n, p) determined by number of trials, n, and probability of success, p.

  6. # of ways to have k successes If we have 1 success in n trials, it could be the first, second, … all the way up to the nth. If we have 2 successes, think about all the ways this could happen: first 2, second 2, first and third,… # of ways to have k successes in n trials:

  7. Combinations 5C2 10C3 8C6 8C2

  8. Binomial Model Binom(n,p) : P(X = x) = nCxpxqn-x p = P(success) q = P(failure) = 1 - p x successes in n trials Expected Value: µ = np Standard Deviation: σ =

  9. Multiple Choice Exam again • 10 questions, each question has 5 choices • Binom(10, 0.2) • Bernoulli trial? • 2 outcomes? • P(success) same for all trials? • Independent trials?/10% condition? • P(X = 7) P(X ≥ 7)

  10. Multiple Choice exam cont’d. Binom(10, 0.2) Expected number of questions correct: Standard deviation:

  11. Using the Normal Model to Estimate the Binomial Model Binom(5, 0.2) Binom(50, 0.2) Figures from Intro Stats, De Veaux

  12. More with the Normal Model Binom(50, 0.2) centered and magnified looks like Normal(10, 2.8) For a large enough number of trials, the Normal model is a close enough approximation Success/Failure condition: Binomial model can be approximated by Normal if we expect at least 10 successes and 10 failures Figure from Intro Stats, De Veaux

  13. Using the Normal Model for a Binomial Model • Suppose an Olympic archer can hit the bull’s-eye 80% of the time. Assume each shot is independent of the others. • If the archer shoots 200 arrows in a competition, • What are the mean and standard deviation of the number of bull’s-eyes? • What is the probability that the archer hits more than 140 bull’s-eyes? Example from HW section, Intro Stats, De Veaux

  14. Colorblindness Example About 8% of males are colorblind. A researcher needs some colorblind subjects for an experiment and begins checking potential subjects. On average, how many men should the researcher expect to check to find one who is colorblind? What’s the probability that the researcher won’t find anyone colorblind in the first 4? Example from HW section, Intro Stats, De Veaux

  15. Genetic trait in frogs example A specific genetic trait is usually found in 1 of every 8 frogs. He collects a dozen frogs. What’s the probability that he finds the trait in: None of the 12 frogs? At least 2 frogs? 3 or 4 frogs? Example from HW section, Intro Stats, De Veaux

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