Binomial Distribution (Calculating)

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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 probability of exactly k successes in a Binomial Distribution?

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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 is the formula for calculating P(X ≥ k) 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 formula for calculating P(X ≤ k) in a Binomial Distribution?

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P(X ≤ k) = Σ P(X = i) for i = 0 to k.

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What does the notation X ~ B(n, p) mean?

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It means that the random variable X follows a Binomial Distribution with n trials and probability p of success.

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

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The expected value E(X) = n * p.

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What is the variance of a Binomial Distribution?

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The variance Var(X) = n * p * (1 - p).

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In the context of Binomial Distribution, what does 'success' mean?

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'Success' refers to the outcome of interest in a binomial experiment, which can vary based on the context (e.g., getting heads in coin flips).

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How do you interpret P(X = k) in a Binomial Distribution?

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P(X = k) represents the probability of obtaining exactly k successes in n independent Bernoulli trials.