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Binomial Settings

Explore the key conditions of binomial settings, where observations are successes or failures, with fixed n, independence, and constant probability of success. Learn through examples like basketball free throws, die rolls, and more.

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Binomial Settings

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  1. Binomial Settings Sec. 8.1A

  2. Binomial Setting (4 conditions) • Each observation is either a “success” or a “failure”—only 2 categories of answers. • Fixed number of observations (n). • All observations are independent. • Probability of success (p) is the same for each observation.

  3. Ex: How many girls? • Only 2 outcomes—boy or girl • Fixed n = 3 children • Independent: Knowing the gender of the one child tells you nothing about the gender of the next child! • Fixed p = .5 for each observation

  4. Ex: Corrine is a 75% free throw shooter. In a key game she shoots 12 free throws and only makes 7 of them. Is this unusual? • Only 2 outcomes—Make or miss • Fixed n = 12 shots • Independent: Knowing the outcome of one basket tells you nothing about the outcome of the next basket (this has actually been studied!) • Fixed p = .75 for each observation

  5. 9 Roll a die 1000 times and count how many of each number. • Only 2 outcomes • Fixed n • Independence • Fixed p • All OK

  6. 10 Roll a die 1000 times and count how many “1”s. • Only 2 outcomes • Fixed n • Independence • Fixed p • All OK

  7. 10 Ex: Deal 10 cards from a shuffled deck and count the red cards. • Only 2 outcomes • Fixed n • Independence • Fixed p • All OK

  8. 10 Ex: Survey blood donors looking for “universal” donors—blood types “O positive” are counted. • Only 2 outcomes • Fixed n • Independence • Fixed p • All OK

  9. 10 Ex: On a 40-question AP Stats exam, count the number of times the answer was “B” • Only 2 outcomes • Fixed n • Independence • Fixed p • All OK

  10. 10 Ex: Test 5 batches of jello mix for solubility, increasing the amount of sugar each time. • Only 2 outcomes • Fixed n • Independence • Fixed p • All OK

  11. 10 Ex: Engineer chooses 10 switches from 10,000. Count the number of bad switches when 10% are bad. • Only 2 outcomes • Fixed n • Independence • Fixed p • All OK

  12. The “10 switches out of 10,000” situation is almost a binomial setting. Technically, choosing one bad switch changes the probability that the next one is bad. BUT choosing 1 switch out of 10,000 does not really change the makeup of the remaining 9999 switches (like choosing 1 card out of 52) so we say the events are independent!

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