Binomial Distributions

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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 (success or failure) across a fixed number of trials.

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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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CCSS.HSS.MD.A.3

CCSS.HSS.MD.A.4

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How do you calculate the probability of getting exactly k successes in n trials?

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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).

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CCSS.HSS.MD.B.6

CCSS.HSS.MD.B.7

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

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'n choose k' (denoted as C(n, k) or nCk) is 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 of a binomial distribution?

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The mean (expected value) of a binomial distribution is calculated as μ = n * p.

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CCSS.HSS.ID.A.4

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What is the variance of a binomial distribution?

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The variance of a binomial distribution is calculated as σ² = n * p * (1 - p).

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What is the standard deviation of a binomial distribution?

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The standard deviation of a binomial distribution is the square root of the variance: σ = √(n * p * (1 - p)).

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In a binomial distribution, what does it mean to have 'at least' k successes?

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'At least' k successes means calculating the probability of getting k or more successes, which can be found by summing the probabilities from k to n.

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How do you find the probability of getting at least k successes?

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To find the probability of getting at least k successes, calculate P(X >= k) = 1 - P(X < k) = 1 - Σ P(X = i) for i = 0 to k-1.

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CCSS.HSS.MD.A.2

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What is the probability of getting exactly 10 heads when flipping a coin 20 times?

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The probability is 17.62%.

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CCSS.HSS.MD.B.6

CCSS.HSS.MD.B.7