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Binomial Distribution quiz

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  • What are the four criteria that define a binomial experiment?
    There must be only two possible outcomes per trial, a fixed number of independent trials, and the probability of success must remain constant for each trial.
  • In a binomial experiment, what does the variable 'n' represent?
    'n' represents the total number of trials conducted in the experiment.
  • How do you calculate the probability of failure (q) in a binomial experiment?
    The probability of failure, q, is calculated as 1 minus the probability of success, p (q = 1 - p).
  • What does the variable 'x' represent in binomial probability problems?
    'x' represents the number of successes in the n trials of the experiment.
  • What is the formula for the probability of getting exactly x successes in n binomial trials?
    The formula is P(X = x) = (n choose x) * p^x * q^(n-x), where (n choose x) is the combination of n items taken x at a time.
  • How do you calculate the expected value (mean) of a binomial distribution?
    The expected value (mean) is calculated as μ = n × p.
  • What is the shortcut formula for the standard deviation of a binomial distribution?
    The standard deviation is σ = √(n × p × q).
  • If a binomial experiment has n = 10 and p = 0.6, what is the expected number of successes?
    The expected number of successes is 10 × 0.6 = 6.
  • Why must trials be independent in a binomial experiment?
    Trials must be independent so that the outcome of one trial does not affect the outcome of another, ensuring constant probability of success.
  • How do you find the probability of a range of successes (e.g., between 0 and 2) in a binomial distribution?
    You add the probabilities for each value in the range: P(X=0) + P(X=1) + P(X=2).
  • What is the complement rule in binomial probability calculations?
    The complement rule states that P(X ≥ k) = 1 - P(X ≤ k-1), allowing you to find probabilities more efficiently.
  • Which calculator function is used to find the probability of exactly x successes in a binomial experiment?
    The binomPDF function is used to find the probability of exactly x successes.
  • Which calculator function is used to find the probability of at most x successes in a binomial experiment?
    The binomCDF function is used to find the probability of at most x successes (P(X ≤ x)).
  • How can you use binomCDF to find the probability of fewer than x successes?
    You use binomCDF with x-1 as the value to find P(X < x), since binomCDF gives P(X ≤ x).
  • What is the relationship between variance and standard deviation in a binomial distribution?
    Variance is the square of the standard deviation, so variance = σ² = n × p × q.