CVHS Binomial Distribution

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What is the Binomial Distribution?

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A probability distribution that summarizes the likelihood that a value will take one of two independent states and is defined by the number of trials (n) and the probability of success (p).

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What are the assumptions of the Binomial Distribution?

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1. Fixed number of trials (n). 2. Each trial is independent. 3. Each trial has only two outcomes (success or failure). 4. The probability of success (p) is constant for each trial.

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What does 'P(X ≥ k)' represent in a Binomial Distribution?

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The probability of getting at least k successes in n trials.

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How do you calculate the probability of exactly k successes in n trials?

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

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What is the meaning of 'n choose k' in probability?

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

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If the probability of success is 0.11, what is the probability of failure?

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The probability of failure is 1 - p = 1 - 0.11 = 0.89.

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What is the probability of getting exactly 2 successes in 10 trials with p = 0.11?

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Use the formula: P(X = 2) = (10 choose 2) * (0.11)^2 * (0.89)^8.

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What does 'at least 2 successes' mean in terms of probability?

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It means calculating the probability of getting 2 or more successes, which can be found by 1 - P(X < 2).

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What is the significance of the mean in a Binomial Distribution?

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The mean (expected value) is calculated as μ = n * p, representing the average number of successes in n trials.

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What is the variance in a Binomial Distribution?

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The variance is calculated as σ² = n * p * (1 - p), indicating the spread of the distribution.