Binomial Distribution

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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 two parameters: n (number of trials) and p (probability of success).

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CCSS.HSS.MD.A.3

CCSS.HSS.MD.A.4

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What are the parameters of a Binomial Distribution?

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The parameters of a Binomial Distribution are n (the number of trials) and p (the probability of success on each trial).

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CCSS.HSS.MD.A.3

CCSS.HSS.MD.A.4

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

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P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where (n choose k) = n! / (k!(n-k)!).

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CCSS.HSS.MD.A.3

CCSS.HSS.MD.A.4

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What does 'n choose k' represent in the Binomial formula?

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'n choose k' represents the number of ways to choose k successes from n trials, calculated as n! / (k!(n-k)!).

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What is the difference between Binomial Probability Density Function (pdf) and Cumulative Distribution Function (cdf)?

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The pdf gives the probability of exactly k successes, while the cdf gives the probability of k or fewer successes.

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CCSS.HSS.MD.A.3

CCSS.HSS.MD.A.4

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If a coin is flipped 20 times, what is the probability of getting exactly 10 heads?

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P(X = 10) = (20 choose 10) * (0.5)^10 * (0.5)^(20-10) = 0.1762.

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CCSS.HSS.MD.B.6

CCSS.HSS.MD.B.7

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How do you calculate the probability of getting at most k successes in a Binomial Distribution?

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Use the cumulative distribution function: P(X ≤ k) = Σ P(X = i) for i = 0 to k.

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What is the probability of getting 6 or more successes in a Binomial Distribution with n=8 and p=0.39?

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P(X ≥ 6) = 1 - P(X ≤ 5) = 0.0439.

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In a Binomial Distribution, what does 'at least' mean?

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'At least' means calculating the probability of k or more successes, which can be found using the cdf: P(X ≥ k) = 1 - P(X ≤ k-1).

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What is the probability of getting exactly 5 successes when 18 trials are conducted with a success probability of 0.25?

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P(X = 5) = (18 choose 5) * (0.25)^5 * (0.75)^(18-5) = 0.199.