Binomial Probability "at least / at most"

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What is the binomial probability formula?

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The binomial probability formula 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 does 'at least' mean in the context of binomial probability?

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'At least' means that the number of successes is greater than or equal to a certain value.

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What does 'at most' mean in the context of binomial probability?

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'At most' means that the number of successes is less than or equal to a certain value.

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How do you calculate the probability of 'at most k' successes in a binomial distribution?

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To calculate the probability of 'at most k' successes, sum the probabilities from 0 to k: P(X ≤ k) = Σ P(X = i) for i = 0 to k.

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How do you calculate the probability of 'at least k' successes in a binomial distribution?

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To calculate the probability of 'at least k' successes, use the complement: P(X ≥ k) = 1 - P(X < k) = 1 - Σ P(X = i) for i = 0 to k-1.

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What is the significance of the parameters n and p in a binomial distribution?

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n is the number of trials, and p is the probability of success on each trial.

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What is the expected value (mean) of a binomial distribution?

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The expected value (mean) of a binomial distribution is E(X) = n * p.

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What is the variance of a binomial distribution?

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The variance of a binomial distribution is Var(X) = n * p * (1 - p).

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In the context of the basketball game question, what is the probability of making no more than 3 free throws?

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The probability of making no more than 3 free throws can be calculated using the binomial probability formula with n = 18 and p = 0.08.

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In the context of the defective balls question, how do you find the probability of at most 4 defective balls?

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Use the binomial probability formula with n = 6 and p = 0.15, summing the probabilities for 0 to 4 defective balls.