Binomial Distributions (4.2)

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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 (success or failure) across a fixed number of trials.

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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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How do you calculate the mean of a binomial distribution?

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The mean (μ) of a binomial distribution is calculated using the formula: μ = n * p.

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How do you calculate the standard deviation of a binomial distribution?

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The standard deviation (σ) of a binomial distribution is calculated using the formula: σ = √(n * p * (1 - p)).

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What is the probability of getting exactly k successes in n trials?

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The probability of getting exactly k successes in n trials is given by the formula: P(X = k) = (n choose k) * p^k * (1 - p)^(n - k).

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What does 'n choose k' mean in the context of binomial distributions?

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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 significance of independent trials in a binomial distribution?

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Independent trials mean that the outcome of one trial does not affect the outcome of another trial, which is a key assumption in binomial distributions.

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What is the probability of success in a binomial distribution?

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The probability of success (p) in a binomial distribution is the likelihood of achieving a success in a single trial.

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What is the probability of failure in a binomial distribution?

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The probability of failure (q) in a binomial distribution is calculated as q = 1 - p.

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If a coin is flipped 10 times, what is the probability of getting exactly 5 heads?

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Using the binomial formula, P(X = 5) = (10 choose 5) * (0.5)^5 * (0.5)^5 = 0.246.