Ch 17 Binomial v. Geometric DIstribution

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What is a Binomial Distribution?

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A Binomial Distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success.

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What is a Geometric Distribution?

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A Geometric Distribution models the number of trials needed to get the first success in a series of independent Bernoulli trials.

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What are the key characteristics of a Binomial Distribution?

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1. Fixed number of trials (n). 2. Two possible outcomes (success or failure). 3. Constant probability of success (p). 4. Independent trials.

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What are the key characteristics of a Geometric Distribution?

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1. Trials continue until the first success. 2. Two possible outcomes (success or failure). 3. Constant probability of success (p). 4. Independent trials.

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In a Binomial Distribution, how is the probability of exactly k successes calculated?

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P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success.

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In a Geometric Distribution, how is the probability of the first success on the k-th trial calculated?

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P(X = k) = (1-p)^(k-1) * p, where p is the probability of success.

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What is the mean of a Binomial Distribution?

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The mean (expected value) of a Binomial Distribution is given by E(X) = n * p.

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What is the mean of a Geometric Distribution?

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The mean (expected value) of a Geometric Distribution is given by E(X) = 1/p.

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How do you determine if a scenario is modeled by a Binomial or Geometric Distribution?

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If the number of trials is fixed and you are counting successes, use Binomial. If you are counting trials until the first success, use Geometric.

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Example of a Binomial Distribution scenario:

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Flipping a coin 10 times and counting the number of heads.