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 outcomes under a given set of parameters.

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CCSS.HSS.MD.A.3

CCSS.HSS.MD.A.4

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What are the key characteristics of a binomial experiment?

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1. The number of trials (n) is fixed. 2. Each trial has only two possible outcomes (success or failure). 3. The trials are independent. 4. The probability of success (p) is the same for each trial.

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What is the formula for calculating 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)!)

Tags

CCSS.HSS.MD.A.3

CCSS.HSS.MD.A.4

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How do you calculate the mean of a binomial distribution?

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Mean (μ) = n * p, where n is the number of trials and p is the probability of success.

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How do you calculate the variance of a binomial distribution?

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Variance (σ²) = n * p * (1 - p), where n is the number of trials and p is the probability of success.

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What is the probability of getting at least k successes 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 does it mean for trials to be independent in a binomial experiment?

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It means the outcome of one trial does not affect the outcome of another trial.

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Can a binomial distribution have a probability of success greater than 1?

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No, the probability of success (p) must be between 0 and 1, inclusive.

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What is the probability of getting exactly 3 successes in 6 trials with a success probability of 0.5?

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P(X = 3) = (6 choose 3) * (0.5)^3 * (0.5)^(6-3) = 0.3125.

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CCSS.HSS.MD.B.6

CCSS.HSS.MD.B.7

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What is the cumulative distribution function (CDF) for a binomial distribution?

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The CDF gives the probability that a binomial random variable is less than or equal to a certain value k.

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CCSS.HSS.ID.A.4