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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What are the key assumptions of a Binomial Distribution?

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1. Fixed number of trials. 2. Each trial is independent. 3. Each trial has only two outcomes (success or failure). 4. The probability of success is constant across trials.

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What is the formula for calculating the probability of exactly k successes in n trials in a Binomial Distribution?

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P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where p is the probability of success.

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What does 'n choose k' mean in the context of Binomial Distribution?

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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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If a fair coin is flipped 10 times, what is the probability of getting exactly 5 heads?

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P(X = 5) = (10 choose 5) * (0.5)^5 * (0.5)^(10-5) = 0.2461 or 24.61%.

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What is the expected value (mean) of a Binomial Distribution?

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The expected value (mean) is calculated as E(X) = n * p, where n is the number of trials and p is the probability of success.

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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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In a Binomial Distribution, what does the term 'success' refer to?

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'Success' refers to the outcome of interest in a trial, which can vary based on the context of the problem.

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What is the probability of rolling a '4' exactly 25 times when rolling a fair die 100 times?

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1.0%.

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Which of the following is NOT an assumption of the Binomial distribution?

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All trials are dependent on each other.