Glossary
BINS (mnemonic)
An acronym (Binary, Independent, Number of trials fixed, Same probability of success) used to remember the four conditions required for a binomial setting.
Example:
Before applying binomial probability, always check the BINS conditions to ensure the scenario fits the model.
Binomial Coefficient (nCx)
Represents the number of distinct ways to choose exactly *x* successes from *n* total trials, without regard to order.
Example:
The binomial coefficient calculates that there are 6 ways to choose 2 items from a set of 4.
Binomial Probability Formula
The mathematical formula P(X = x) = nCx * p^x * (1-p)^(n-x) used to calculate the probability of exactly *x* successes in *n* trials.
Example:
Using the binomial probability formula allows you to manually calculate the probability of getting exactly 2 heads in 4 coin flips.
Binomial Random Variable
A random variable (X) that counts the number of successes in a fixed number of independent trials.
Example:
The number of students who pass a specific AP exam out of a random sample of 50 students is a binomial random variable.
Constant Probability of Success (p)
A condition for a binomial setting where the probability of achieving a 'success' remains the same for every trial.
Example:
If a free-throw shooter consistently makes 75% of their shots, their constant probability of success p is 0.75 for each attempt.
Empirical Probability
Probability estimated by simulating a random event many times and using the observed frequencies.
Example:
If a baseball player gets 30 hits in 100 at-bats, their empirical probability of getting a hit is 0.30.
Fixed Number of Trials (n)
A condition for a binomial setting where the random event is performed a set, predetermined number of times.
Example:
When flipping a coin 20 times, the fixed number of trials n is 20.
Independent Trials
A condition for a binomial setting where the outcome of one trial does not influence the outcome of any other trial.
Example:
Each flip of a fair coin is an independent trial because the result of one flip doesn't affect the next.
Interpretation in Context
The process of explaining the meaning of a statistical result or probability in clear, non-technical language, relating it back to the original problem scenario.
Example:
Stating 'There is a 12% chance that exactly 3 out of 10 surveyed people will like the new snack' is an example of interpretation in context.
Probability Distribution
A map of possible outcomes for a random event that shows the likelihood of each outcome.
Example:
The probability distribution for rolling a fair six-sided die assigns a 1/6 chance to each number from 1 to 6.
Probability of Failure (1-p)
The specific likelihood of the undesired outcome (failure) occurring in a single trial within a binomial setting.
Example:
If the probability of a new product being successful is 0.80, then the probability of failure (1-p) is 0.20.
Probability of Success (p)
The specific likelihood of the desired outcome (success) occurring in a single trial within a binomial setting.
Example:
If 15% of online ads lead to a click, then the probability of success p for a single ad is 0.15.
Theoretical Probability
Probability calculated using the rules of probability, suitable for situations with clear-cut probabilities.
Example:
Determining the chance of drawing an ace from a standard deck of cards (4/52) is an example of theoretical probability.
Two Outcomes (Success or Failure)
A condition for a binomial setting where each trial can only result in one of two possible results, typically labeled as 'success' or 'failure'.
Example:
In a quality control check, each product is either 'defective' (failure) or 'not defective' (success), fulfilling the two outcomes condition.
binomCDF
A calculator function used to find the cumulative probability of obtaining *x or fewer* successes in a binomial distribution.
Example:
To find the probability of a customer making 3 or fewer purchases from a series of 10 marketing emails, you would use binomCDF.
binomPDF
A calculator function used to find the probability of obtaining *exactly* a specified number of successes in a binomial distribution.
Example:
To find the probability of a student guessing exactly 5 questions correctly on a 20-question true/false quiz, you would use binomPDF.