Parameters for a Binomial Distribution Chapter Notes | AP Statistics - Grade 9 PDF Download

In order to use the binomial distribution to model a random event, the event must meet the following four conditions:

• Binary: The possible outcomes of each trial can be classified as a success or a failure.
• Independent: Trials must be independent. Knowing the outcome of one trial must not inform the outcome of any other trial.
• Fixed number: The number of trials n must be fixed in advance.
• Same probability: There must be the same probability of success p on each trial.

(A mnemonic device that might help is BINS: binary, independent, number, same probability!)
If any of these conditions is not met, then the event cannot be modeled using the binomial distribution. For example, if you want to model the number of heads in 10 flips of a biased coin with varying probabilities of heads, then the event cannot be modeled using the binomial distribution.

Describing Mean and Standard Deviation of Binomial Variables

• Mean: The mean (expected value) of a binomial random variable X, representing the number of successes in n independent trials with probability of success p, is given by: mean = E(X) = n * p
• Standard deviation: The standard deviation of a binomial random variable X is given by: standard deviation = σ_X = sqrt(n * p * (1 - p))

Binomial Distributions in Statistical Sampling

Key Terms to Review

• 10% Condition: A guideline ensuring that the sample size taken from a population is small enough relative to the population size, typically indicating that the sample size should be less than 10% of the total population.
• Binomial Distribution: A probability distribution summarizing the likelihood of a given number of successes in a fixed number of independent Bernoulli trials.
• Expected Value: Represents the average outcome of a random variable based on its possible values, weighted by the likelihood of occurrence.
• Failure: The outcome that is not classified as a success in a binomial distribution.
• Mean: A measure of central tendency that represents the average value of a set of numbers.
• Probability of Success: The likelihood that a given outcome will occur in a statistical experiment.
• Random Variable: A numerical outcome of a random process, classified into discrete and continuous types.
• Random Sample: A subset of individuals chosen from a larger population where each individual has an equal chance of being selected.
• Standard Deviation: A measure of the amount of variation or dispersion in a set of values.