Binomial and Geometric Distributions

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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 defined by the number of trials (n) and the probability of success (p).

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What are the assumptions of a Binomial Distribution?

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1. Fixed number of trials (n). 2. Each trial is independent. 3. Each trial has only two outcomes (success or failure). 4. The probability of success (p) is constant for each trial.

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How do you calculate the mean of a Binomial Distribution?

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Mean (μ) = n * p, where n is the number of trials and p is the probability of success.

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How do you calculate the standard deviation of a Binomial Distribution?

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Standard Deviation (σ) = √(n * p * (1 - p)).

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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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How do you calculate the mean of a Geometric Distribution?

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Mean (μ) = 1/p, where p is the probability of success.

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How do you calculate the standard deviation of a Geometric Distribution?

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Standard Deviation (σ) = √((1 - p) / p²).

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What is the probability of getting exactly k successes in n trials in a Binomial Distribution?

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P(X = k) = (n choose k) * p^k * (1 - p)^(n - k).

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What is the probability of getting at least k successes in n trials in a Binomial Distribution?

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P(X ≥ k) = 1 - P(X < k) = 1 - Σ[P(X = i)] for i = 0 to k-1.

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What is the difference between Binomial and Geometric Distributions?

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Binomial Distribution counts the number of successes in a fixed number of trials, while Geometric Distribution counts the number of trials until the first success.