Binomial 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 two parameters: n (number of trials) and p (probability of success).

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What are the parameters of a Binomial Distribution?

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The parameters of a Binomial Distribution are n (the number of trials) and p (the probability of success on each trial).

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How do you calculate the mean of a Binomial Distribution?

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The mean (μ) of a Binomial Distribution is calculated using the formula: μ = n * p.

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

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The standard deviation (σ) of a Binomial Distribution is calculated using the formula: σ = √(n * p * (1 - p)).

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What is the probability of getting at least one success in a Binomial Distribution?

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The probability of getting at least one success is calculated as: P(X ≥ 1) = 1 - P(X = 0), where P(X = 0) is the probability of zero successes.

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What is the formula for calculating the probability of exactly k successes in a Binomial Distribution?

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The formula is: P(X = k) = (n choose k) * p^k * (1 - p)^(n - k), where (n choose k) = n! / (k!(n-k)!).

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If n = 20 and p = 0.1, what is the probability of getting exactly 2 successes?

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Using the formula: P(X = 2) = (20 choose 2) * (0.1)^2 * (0.9)^(20-2) = 0.1937.

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What does 'at most k successes' mean in a Binomial Distribution?

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'At most k successes' means the probability of getting k or fewer successes, calculated as P(X ≤ k).

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How do you interpret the standard deviation in a Binomial Distribution?

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The standard deviation indicates the variability or spread of the distribution; a larger standard deviation means more variability in the number of successes.

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What is the relationship between Binomial Distribution and the Normal Distribution?

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For large n, the Binomial Distribution can be approximated by the Normal Distribution, especially when both np and n(1-p) are greater than 5.