Ch 17 Binomial v. Geometric DIstribution

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What is a Binomial Distribution?

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A Binomial Distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success.

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What is a Geometric Distribution?

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A Geometric Distribution models the number of trials needed to get the first success in a series of independent Bernoulli trials.

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When do you use Binomial Distribution?

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Use Binomial Distribution when you have a fixed number of trials, each with two possible outcomes (success or failure), and you want to find the probability of a certain number of successes.

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When do you use Geometric Distribution?

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Use Geometric Distribution when you are interested in the number of trials until the first success occurs.

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What is the formula for Binomial Probability?

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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 Geometric Probability?

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P(X = k) = (1-p)^(k-1) * p, where k is the trial on which the first success occurs and p is the probability of success.

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Example of Binomial Distribution: A basketball player makes 70% of their free throws. What is the probability of making exactly 5 out of 8 free throws?

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Use the Binomial formula: P(X = 5) = (8 choose 5) * (0.7)^5 * (0.3)^3.

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Example of Geometric Distribution: A printer produces labels with an 80% error-free rate. What is the probability that the first defective label occurs on the 6th attempt?

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Use the Geometric formula: P(X = 6) = (0.2)^(5) * (0.8).

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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.

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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.