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 characterized by the number of trials, the probability of success in each trial, and the number of successes.

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CCSS.HSS.MD.A.4

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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 getting exactly k successes in n trials in a binomial distribution?

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The probability of getting exactly k successes in n trials is calculated using the formula: P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where (n choose k) = n! / (k!(n-k)!).

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CCSS.HSS.MD.A.3

CCSS.HSS.MD.A.4

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What does 'n choose k' mean in the context of binomial distribution?

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

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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 given by the formula: E(X) = n * p.

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CCSS.HSS.MD.A.2

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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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In the context of the binomial distribution, what does 'success' mean?

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'Success' refers to the outcome of interest in a binomial experiment, which is the event we are measuring the probability of.

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How do you determine the probability of getting at least k successes in a binomial distribution?

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To determine the probability of getting at least k successes, you can calculate 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 relationship between binomial distribution and Bernoulli trials?

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A binomial distribution is the sum of a fixed number of independent Bernoulli trials, where each trial has two possible outcomes: success or failure.

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How can the binomial distribution be applied in real-world scenarios?

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The binomial distribution can be applied in various scenarios such as quality control, medical trials, and predicting outcomes in sports.