Binomial Experiments

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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) It consists of n identical trials. 2) Each trial has two possible outcomes: success or failure. 3) The probability of success (p) is the same for each trial. 4) The trials are independent.

Front

Identify the parameters of a binomial experiment: n, p, q, and x.

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n = number of trials, p = probability of success, q = probability of failure (q = 1 - p), x = number of successes.

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In a binomial experiment, what does the random variable represent?

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The random variable in a binomial experiment represents the number of successes in n trials.

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What is the formula for calculating the probability of exactly x successes in a binomial experiment?

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P(X = x) = (n choose x) * (p^x) * (q^(n-x)) where (n choose x) = n! / (x!(n-x)!)

Front

If a fair coin is flipped 12 times, what is the probability of getting exactly 6 heads?

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P(X = 6) = 0.23.

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How many outcomes are there in a binomial experiment?

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There are 2 outcomes in a binomial experiment: success and failure.

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What is the probability of getting at least one success in a binomial experiment?

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P(at least 1 success) = 1 - P(no successes) = 1 - (q^n).

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In a binomial experiment, if n = 7 and p = 0.7, what is the probability of getting exactly 2 successes?

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Use the binomial formula: P(X = 2) = (7 choose 2) * (0.7^2) * (0.3^5).

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What does it mean if a situation does not describe a binomial experiment?

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It means that one or more of the properties of a binomial experiment are not satisfied, such as having more than two outcomes or dependent trials.

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In a state lottery, if you choose 6 numbers from 1 to 40, what is the probability of matching exactly 3 numbers?

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Use the binomial formula with n = 6, p = 6/40, and calculate P(X = 3).