MTH 232
Section 14.2
Applications of Counting Principles to
Probability
Overview
• Recall that the probability of an event E is
given by:
n( E )
Pr( E ) 
n( S )
Where n(E) is the number of ways E can occur
and n(S) is the number of elements in the
sample space S.
Counting
• For small experiments, S can usually be listed and
elements in E can be identified.
• However, for larger experiments it becomes
necessary to calculate n(S) and n(E) using other
techniques:
1. Tree diagrams
2. Venn diagrams
3. The multiplication principle of counting
4. The addition principle of counting
Tree Diagrams
• Comprised of nodes and branches.
• Each node indicates a choice.
• The branch that leads to that choice is labeled
with the probability of that choice.
• Tree diagrams are good for small experiments
with multiple stages.
An Example
• A bag contains five white and three black
balls. Two balls are drawn without
replacement. Construct a probability tree to
determine the following probabilities:
1. Both balls are white.
2. Both balls are black.
3. At least one black ball is drawn.
Venn Diagrams
• Previously covered in MTH 231; better used with two
sets with overlapping elements.
• Example: A total of 40 students are either Shelton State
Ambassadors or members of Phi Theta Kappa (some
are in both organizations). If 25 students are
Ambassadors and 18 students are in Phi Theta Kappa:
1. How many students are in both organizations?
2. How many students are only Ambassadors?
3. What is the probability that a randomly selected
student is only in Phi Theta Kappa?
4. What is the probability that a randomly selected
student is in both organizations?
The Multiplication Principle
• Best used with a multi-stage, or multi-step
experiment.
• Find the number of ways each stage, or step,
can occur.
• Multiply those numbers together.
• Consider whether repetition is allowed or not
allowed.
Examples
• License plates (find the number of possible
license plates in Tuscaloosa County. Is it the
same as the number of possible license plates
in Jefferson County?)
• Passwords based on a choice of numbers or
letters (how many 4 digit passwords are
possible? How many are possible is repetition
of digits is not allowed?)
The Addition Principle
• Used to count outcomes (with or without
overlap).
• Can be used with simple experiments
involving cards and dice (where the number of
elements in the sample space is already
known).
Examples
• What is the probability of drawing a prime
number from a standard deck of playing
cards?
• What is the probability of rolling a prime
number if rolling two dice?
Modified Homework
• 16: a, b, and c only
• 17: a and b only