Binomial Probability

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What is Binomial Probability?

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Binomial Probability refers to the probability of obtaining a fixed number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success.

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What are the assumptions of the Binomial distribution?

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1. The number of trials is fixed. 2. Each trial is independent. 3. Each trial has only two possible outcomes (success or failure). 4. The probability of success is constant across trials.

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

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P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where p is the probability of success.

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What does 'n choose k' mean in the context of Binomial Probability?

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'n choose k' (denoted as C(n, k) or nCk) is the number of ways to choose k successes from n trials, calculated as n! / (k!(n-k)!).

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

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17.62%.

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What is the probability that a quarterback with a 70% passing percentage does NOT complete 8 of his 20 passes?

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11.44%.

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In a survey, if 25% of pet owners have their pets bathed professionally, what is the probability that exactly 5 out of 18 pet owners do so?

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0.199.

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What is the probability of getting exactly 6 heads when a coin is tossed 10 times?

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20.51%.

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What is a Bernoulli trial?

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A Bernoulli trial is a random experiment where there are only two possible outcomes: success or failure.

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What is the significance of independence in Binomial Probability?

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Independence means the outcome of one trial does not affect the outcome of another, which is crucial for applying the Binomial distribution.