Binomial & Geometric Distribution Practice

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What is the Geometric Distribution?

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A probability distribution that models the number of trials needed to achieve the first success in a series of independent Bernoulli trials.

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What is the Binomial Distribution?

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A probability distribution that models the number of successes in a fixed number of independent Bernoulli trials.

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When would you use the Geometric Distribution?

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When you want to find the probability of the first success occurring on a specific trial.

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When would you use the Binomial Distribution?

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When you want to find the probability of a certain number of successes in a fixed number of trials.

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

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P(X = k) = C(n, k) * p^k * (1-p)^(n-k), where C(n, k) is the binomial coefficient.

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What does 'C(n, k)' represent in the Binomial formula?

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It represents the number of combinations of n items taken k at a time.

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

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

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How do you calculate the probability of not getting a success in a Geometric Distribution?

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

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What is the probability of rolling a 4 exactly 25 times in 100 rolls of a die?

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Use the Binomial formula: P(X = 25) = C(100, 25) * (1/6)^25 * (5/6)^(75).