Introduction to Binomial Distributions (College Board AP® Statistics)

Conditions for binomial distributions

What is a binomial distribution?

• A discrete random variable follows a binomial distribution if it counts the number of successes when an experiment satisfies the following conditions:

  • there are a fixed number of repeated trials ()

  • the trials are independent of each other

  • there are exactly two outcomes of each trial (success or failure)

  • the probability of success () is constant for each trial

• The notation for a binomial random variable is

  • is the number of trials

  • is the probability of success

    • is the probability of failure

• can take the integer values 0, 1, 2, ...,

Modeling with binomial distributions

What can be modeled by binomial distributions?

• A binomial distribution can model any variable that satisfies the four conditions

  • e.g. the number of times a coin lands on heads when flipped 20 times

    • The coin is flipped 20 times

    • Each coin flip is independent of each other

    • A success is landing on heads and a failure is landing on tails

    • The probability of landing on heads is constant as the same coin is used each time

• Binomial models can sometimes still be used when it seems like there are more than two outcomes

  • e.g. the number of times a six-sided dice lands on a six when rolled 20 times

    • Although the dice could land on one of six numbers, for this experiment there are only two outcomes: landing on a six (success) or not landing on a six (failure)

  • Binomial distributions are good models in these situations provided that a successful outcome can be clearly defined

• Binomial models can sometimes still be used when the independence condition is not fulfilled

  • e.g. the number of people in a sample of 30 who are unemployed when it is known that 50 people are unemployed out of a population of 1,000 people

    • The probability that the first person sampled is unemployed is

    • The probability that the next person sampled is unemployed depends on whether the first person is unemployed or not ( or )

  • Binomial distributions are good models in these situations provided the size of the population is at least 10 times the size of the sample

What cannot be modeled by binomial distributions?

• Binomial models are not suitable if:

  • there is not a fixed number of trials

    • e.g. the number of emails received in an hour could be infinite

  • the trials are not independent

    • e.g. the number of blue tokens obtained when five tokens are taken from a bag containing 6 blue tokens and 4 red tokens

  • there are more than two outcomes

    • e.g. the number a dice lands on when rolled

  • the probability of success changes

    • e.g. the number of times an archer hits a target out of ten shots if they take a step closer to the target after each shot

Properties of binomial distributions

What is the shape of a binomial distribution?

• The value of in a binomial distribution , determines the skewness of the distribution

Value of	Skewness
	Positive skew
	Symmetrical
	Negative skew

How do I calculate the mean and standard deviation of a binomial distribution?

• If then

  • the mean is

  • the standard deviation is

    • These formulas are given in the exam