Binomial v. Geometric DIstribution

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What is a Binomial Distribution?

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A Binomial Distribution is a probability distribution that summarizes the likelihood that a value will take one of two independent states and is characterized by a fixed number of trials, each with the same probability of success.

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What is a Geometric Distribution?

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A Geometric Distribution is a probability distribution that models the number of trials needed to achieve the first success in a series of independent Bernoulli trials, each with the same probability of success.

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When is a Binomial Distribution used?

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A Binomial Distribution is used when there are a fixed number of trials, two possible outcomes (success or failure), and the probability of success is constant across trials.

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When is a Geometric Distribution used?

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A Geometric Distribution is used when we are interested in the number of trials until the first success occurs, with each trial being independent and having the same probability of success.

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What is the formula for the Binomial Probability?

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The formula for Binomial Probability is 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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What is the formula for the Geometric Probability?

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The formula for Geometric Probability is P(X=k) = (1-p)^(k-1) * p, where k is the number of trials until the first success and p is the probability of success.

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Example of a Binomial Distribution scenario.

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Example: Flipping a coin 10 times and counting the number of heads (successes) is a Binomial Distribution scenario.

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Example of a Geometric Distribution scenario.

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Example: Rolling a die until a 6 appears is a Geometric Distribution scenario.

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What is the mean of a Binomial Distribution?

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The mean of a Binomial Distribution is given by the formula μ = n * p, where n is the number of trials and p is the probability of success.

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What is the mean of a Geometric Distribution?

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The mean of a Geometric Distribution is given by the formula μ = 1/p, where p is the probability of success.