This document defines and provides examples of binomial probability distributions. It explains that a binomial probability distribution is a table that lists the possible numbers of successes from n trials paired with each outcome's probability. It also outlines how to calculate the mean, variance, and standard deviation of a binomial distribution. Finally, it demonstrates how to use the distribution table to calculate various probabilities, such as the probability of exactly, at least, more than, or within a range of successes.

1. Introduction
Binomial Probability Distributions

2. Definition
A table of all the possible numbers
of successes for n number of trials
paired with the probability of that
number of success.

3. Example
Number of
successes, x
Probability
of each x

4. StudyTip
The sum of the
probabilities must
equal 1

5. Graph
The number of successes is
the independent variable, and
graphed on the horizontal axis
The probability is the dependent variable,
and graphed on the vertical axis

6. Graph The graphs
always make
this “bell” like
shape

7. Formula Mean of the distribution:
 = np
Variance of the distribution:
2 = npq
Standard deviation of the distribution:
 = npq

8. Example
Notice that the
mean lies within
the most likely
outcomes. It is the
expected value
Notice that the
mean lies at the
peak of the graph

9. Example
The standard deviation tells us what is
normal. Any event more than 2 standard
deviations from the mean is considered
highly unlikely. 2 standard deviations
from the mean covers the bell part of
the graph

10. ProblemSolving
Use the distribution table to answer
questions like:
What is the probability of exactly 3
successes?
What is the probability of at least 3
successes?
What is the probability of more than 3
successes?
What is the probability 1 to 4
successes inclusive?

11. Example
What is the probability
of exactly 3 successes?
The answer is the
probability associated
with x = 3
P(x =3) = .298571

12. Example
What is the probability
of at least 3
successes?
The answer is the sum of
the probabilities for x=3
to x=12
P(x  3) = .298571

13. Example
What is the probability of
more than 3 successes?
The answer is the sum of
the probabilities for x=4
to x=12
P(x > 3) = .111126

14. Example
What is the probability 1
to 4 successes inclusive?
The answer is the sum of
the probabilities from x=1
to x=4
P(1  x  4) = .845512