Binomial Vs Geometric Distributions

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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, two possible outcomes (success or failure), and a constant probability of success.

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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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CCSS.HSS.MD.A.3

CCSS.HSS.MD.A.4

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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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CCSS.HSS.MD.A.3

CCSS.HSS.MD.A.4

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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 success is 0.2, what is the probability that the first success occurs on the 5th trial?

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Use the Geometric formula: P(X=5) = (0.8)^(4) * (0.2).

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CCSS.HSS.MD.B.6

CCSS.HSS.MD.B.7

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

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CCSS.HSS.MD.A.2