2.10 The Binomial Distribution
- Syllabus
- 2026
- Topic
- 2.10
- Level
- —
Learning objectives
2.10A—Justify why a random variable is or is not a binomial random variableJustify why a random variable is or is not a binomial random variable.• A binomial random variable, X, is a discrete random variable that counts the number of successes in repeated independent trials, n, that have only two possible outcomes (success or failure), with the probability of success p and the probability of failure 1−p .2.10B—Calculate the mean and standard deviation for a binomial distributionCalculate the mean and standard deviation for a binomial distribution.• If a random variable is binomial, its mean, µX, is np and its standard deviation, σX, is np(1-p .)2.10C—Interpret the mean, standard deviation, and probabilities for a binomial distributionInterpret the mean, standard deviation, and probabilities for a binomial distribution.• The mean, standard deviation, and probabilities for a binomial distribution should be interpreted in context.2.10D—Estimate probabilities of binomial random variables using data from a simulationEstimate probabilities of binomial random variables using data from a simulation.• A probability distribution can be constructed using the rules of probability or estimated with a simulation. Probability, Random Variables, and Probability Distributions UNIT 22.10E—Calculate probabilities for a binomial distributionCalculate probabilities for a binomial distribution.• The probability that a binomial random variable, X, has exactly x successes for n independent trials, when the probability of success is p, is calculated as ( n) PX() | |= x = ppx - nx | x| ()1 - ( ) , x = 012,, ,, ... n. This is called the binomial probability function. Probability, Random Variables, and Probability Distributions UNIT 2 74
2.10.A—Justify why a random variable is or is not a binomial random variable
Justify why a random variable is or is not a binomial random variable.
- A binomial random variable, X, is a discrete random variable that counts the number of successes in repeated independent trials, n, that have only two possible outcomes (success or failure), with the probability of success p and the probability of failure 1−p .
2.10.B—Calculate the mean and standard deviation for a binomial distribution
Calculate the mean and standard deviation for a binomial distribution.
- If a random variable is binomial, its mean, µX, is np and its standard deviation, σX, is np(1-p .)
2.10.C—Interpret the mean, standard deviation, and probabilities for a binomial distribution
Interpret the mean, standard deviation, and probabilities for a binomial distribution.
- The mean, standard deviation, and probabilities for a binomial distribution should be interpreted in context.
2.10.D—Estimate probabilities of binomial random variables using data from a simulation
Estimate probabilities of binomial random variables using data from a simulation.
- A probability distribution can be constructed using the rules of probability or estimated with a simulation. Probability, Random Variables, and Probability Distributions UNIT 2
2.10.E—Calculate probabilities for a binomial distribution
Calculate probabilities for a binomial distribution.
- The probability that a binomial random variable, X, has exactly x successes for n independent trials, when the probability of success is p, is calculated as ( n) PX() | |= x = ppx - nx | x| ()1 - ( ) , x = 012,, ,, ... n. This is called the binomial probability function. Probability, Random Variables, and Probability Distributions UNIT 2 74