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 are n (the number of trials) and p (the probability of success on each trial).

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What does X ~ B(n, p) represent?

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

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How do you calculate the probability of exactly k successes in a Binomial Distribution?

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Use the formula P(X = k) = (n choose k) * p^k * (1-p)^(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 cumulative distribution function (CDF) in the context of Binomial Distribution?

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The CDF gives the probability that the random variable X is less than or equal to a certain value k, denoted as P(X ≤ k).

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What is the mean of a Binomial Distribution?

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The mean (expected value) is calculated as E(X) = n * p.

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What is the variance of a Binomial Distribution?

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

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If X ~ B(20, 0.5), what is the probability of getting at least 7 successes?

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P(X ≥ 7) = 0.94.

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If X is the number of heads when a fair coin is flipped 12 times, how do you find P(X ≤ 6)?

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Use the CDF to find P(X ≤ 6) which equals 0.61.