Binomial Distributions

Front

What is a binomial random variable?

Back

A binomial random variable is a variable that counts the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success.

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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

Front

How do you calculate the mean of a binomial distribution?

Back

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

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

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The formula is P(X = k) = (n choose k) * p^k * (1 - p)^(n - k).

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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) represents the number of ways to choose k successes from n trials.

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What is the standard deviation of a binomial distribution?

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The standard deviation is calculated as σ = √(n * p * (1 - p)).

Front

If the probability of success is 0.32 and there are 75 trials, what is the standard deviation?

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The standard deviation is approximately 4.24.

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What is the probability of getting at least k successes in n trials?

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To find the probability of getting at least k successes, sum the probabilities of getting k, k+1, ..., n successes.

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How do you find the probability of getting exactly 3 correct answers when guessing on 6 multiple choice questions with 4 options?

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Use the binomial formula with n=6, k=3, and p=0.25 (1/4). The probability is approximately 0.132.