Binomial Distribution - Math 4

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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 are the assumptions of a Binomial Distribution?

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1. The number of trials (n) is fixed. 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 Binomial Random Variable?

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A Binomial Random Variable is a variable that counts 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 the formula for calculating the probability of exactly k successes in n trials?

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P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where (n choose k) = n! / (k!(n-k)!)

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What does 'n choose k' represent in the Binomial formula?

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'n choose k' represents the number of ways to choose k successes from n trials, calculated as n! / (k!(n-k)!).

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

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

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What is the variance of a Binomial Distribution?

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The variance of a Binomial Distribution is given by the formula Var(X) = n * p * (1 - p).

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How do you determine if a situation is Binomial?

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A situation is Binomial if it meets these criteria: fixed number of trials, only two outcomes (success/failure), independent trials, and constant probability of success.

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Is rolling a die ten times and recording the number shown a Binomial experiment?

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No, it is not a Binomial experiment because the outcomes are not limited to two categories.

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Is rolling a die ten times and counting how many threes are rolled a Binomial experiment?

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Yes, it is a Binomial experiment because it has a fixed number of trials, two outcomes (rolling a three or not), and independent trials.