Binomial Vs Geometric 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 states and is defined by the number of trials (n), the number of successes (k), and the probability of success (p).

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What is a Geometric Distribution?

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A Geometric Distribution is a probability distribution that models the number of trials needed to achieve the first success in a series of independent Bernoulli trials, characterized by the probability of success (p).

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When is a Binomial Distribution used?

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A Binomial Distribution is used when there are a fixed number of trials, each trial has two possible outcomes (success or failure), and the probability of success is constant across trials.

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When is a Geometric Distribution used?

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A Geometric Distribution is used when we are interested in the number of trials until the first success occurs, with each trial being independent and having the same probability of success.

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What is the formula for the Binomial Probability?

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The formula for the Binomial Probability is P(X=k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success.

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What is the formula for the Geometric Probability?

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The formula for the Geometric Probability is P(X=k) = (1-p)^(k-1) * p, where k is the number of trials until the first success and p is the probability of success.

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Example of Binomial Distribution: If a coin is flipped 10 times, what is the probability of getting exactly 4 heads?

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Use the Binomial formula: P(X=4) = (10 choose 4) * (0.5)^4 * (0.5)^(10-4).

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Example of Geometric Distribution: If the probability of rolling a 6 on a die is 1/6, what is the probability that the first 6 appears on the 3rd roll?

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Use the Geometric formula: P(X=3) = (5/6)^(3-1) * (1/6).

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What does 'n choose k' mean in Binomial Distribution?

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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 and is calculated as n! / (k!(n-k)!).

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What is the expected value of a Binomial Distribution?

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