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

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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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The probability of exactly k successes in a Binomial Distribution is calculated using the formula: P(X = k) = (n choose k) * p^k * (1-p)^(n-k).

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What does P(X ≤ k) represent in a Binomial Distribution?

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P(X ≤ k) represents the cumulative probability of getting k or fewer successes in n trials.

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What does P(X ≥ k) represent in a Binomial Distribution?

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P(X ≥ k) represents the probability of getting k or more successes in n trials.

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How do you find the mean of a Binomial Distribution?

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

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How do you find the variance of a Binomial Distribution?

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The variance of a Binomial Distribution is calculated using the formula: σ² = n * p * (1 - p).

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What is the significance of the 'n choose k' term in the Binomial formula?

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'n choose k' (denoted as C(n, k) or nCk) represents the number of ways to choose k successes from n trials.

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If a coin is flipped 12 times, what is the probability of getting 6 heads?

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To find P(X = 6) for X ~ B(12, 0.5), use the Binomial formula: P(X = 6) = (12 choose 6) * (0.5)^6 * (0.5)^6.

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What is the probability of getting exactly 3 correct answers when guessing on 6 multiple choice questions with 4 options each?

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Use the Binomial formula: P(X = 3) = (6 choose 3) * (0.25)^3 * (0.75)^3.