Binomial and 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 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 possible outcomes (success or failure). 4. The probability of success (p) is constant for each trial.

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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, characterized by the probability of success (p).

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What is the expected value (mean) of a Binomial Distribution?

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The expected value (mean) of a Binomial Distribution is calculated as E(X) = 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 variance of a Binomial Distribution?

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The variance of a Binomial Distribution is calculated as Var(X) = n * p * (1 - p).

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

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The probability is given by the formula P(X = k) = (n choose k) * p^k * (1 - p)^(n - k).

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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.

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What does it mean if a distribution is a probability distribution?

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A distribution is a probability distribution if all probabilities are between 0 and 1, and the sum of all probabilities equals 1.

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What is the standard deviation of a Binomial Distribution?

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The standard deviation of a Binomial Distribution is calculated as SD(X) = sqrt(n * p * (1 - p)).

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What is the probability of exactly 2 successes in 20 trials with a success probability of 0.11?

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Using the Binomial formula, P(X = 2) = (20 choose 2) * (0.11)^2 * (0.89)^(20-2) = 0.282.