4.6 Practice

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What is a binomial experiment?

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A binomial experiment is a statistical experiment that has the following properties: 1) There are a fixed number of trials, 2) Each trial has two possible outcomes (success or failure), 3) The trials are independent, and 4) The probability of success is the same for each trial.

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What is the formula for calculating the probability of getting exactly k successes in n trials in a binomial distribution?

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The formula is P(X = k) = C(n, k) * (p^k) * (q^(n-k)), where C(n, k) is the binomial coefficient, p is the probability of success, q is the probability of failure (1-p), n is the number of trials, and k is the number of successes.

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What does C(n, k) represent in binomial probability?

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C(n, k) represents the number of combinations of n items taken k at a time, calculated as n! / (k!(n-k)!), where '!' denotes factorial.

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What is the probability of success in a binomial experiment?

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The probability of success is the likelihood of achieving a successful outcome in a single trial of a binomial experiment, denoted as 'p'.

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What is the probability of failure in a binomial experiment?

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The probability of failure is the likelihood of not achieving a successful outcome in a single trial, denoted as 'q', where q = 1 - p.

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If a test has 10 questions and a student has a 70% chance of answering each question correctly, what is the probability of answering exactly 7 questions correctly?

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P(X = 7) = C(10, 7) * (0.7^7) * (0.3^3) = 0.266.

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What is the expected value in a binomial distribution?

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The expected value (mean) is calculated as E(X) = n * p, where n is the number of trials and p is the probability of success.

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What is the variance in a binomial distribution?

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The variance is calculated as Var(X) = n * p * q, where n is the number of trials, p is the probability of success, and q is the probability of failure.

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What is the standard deviation in a binomial distribution?

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The standard deviation is the square root of the variance, calculated as SD(X) = √(n * p * q).

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What is the difference between independent and dependent trials in probability?

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Independent trials are those where the outcome of one trial does not affect the outcome of another, while dependent trials are those where the outcome of one trial influences the outcome of another.