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Chapter 4: Elementary Probability Theory

Section Objectives

• By the end of this section, students should be able to:

  • Organize outcomes in a sample space using tree diagrams.

  • Compute the number of ordered arrangements of outcomes using permutations.

  • Compute the number of (non-ordered) groupings of outcomes using combinations.

  • Explain how counting techniques relate to probability in everyday life.

Tree Diagrams and Counting Techniques

Definition of Tree Diagrams

• Tree Diagram: A graphical representation used to display all possible outcomes in an experiment consisting of a series of events. It helps in organizing and visualizing the outcome space clearly.

Multiplication Rule of Counting

• Multiplication Rule of Counting: A principle that states when calculating the total number of possible outcomes in an experiment with multiple steps, the total number of outcomes is the product of the number of choices at each step.

Example Scenario: Outfits

• Scenario: Suppose you own three shirts and four pairs of pants.

  • Question: How many different outfits can you create?

  • Solution: Each shirt has four pants choices, therefore:

  • Total outfit combinations = $3 imes 4 = 12$

  • This illustrates the multiplication rule of counting.

Formal Definition

• If there is a series of events $E<em>1$ through $E</em>m$ where:

  • $n<em>1$ is the number of possible outcomes for event $E</em>1$,

  • $n<em>2$ is the number of possible outcomes for event $E</em>2$,

  • …

  • $n<em>m$ is the number of possible outcomes for event $E</em>m$,

  • Then the total number of possible outcomes for the series is given by:

  • $n<em>1 imes n</em>2 imes … imes n_m$

Implementing the Multiplication Rule - Example 1

• Scenario: Jacqueline wants to take:

  • 1 course in psychology (2 sections available)

  • 1 course in physiology (2 sections available)

  • 1 course in Spanish (3 sections available)

  • Question: How many different class schedules can Jacqueline choose from?

  • Solution:

  • By the multiplication rule:

  • Total class schedules possible = $2 imes 2 imes 3 = 12$

Additional Concepts: Outcomes and Dependencies

Example Scenario: Colored Discs

• Scenario: 5 discs: 3 red, 2 blue. Draw one, note the color, set aside, then draw a second.

  • Question: Identify the possible outcomes of the experiment.

  • Dependent Events: The outcome of the second draw is dependent on the first since the initial disc is not replaced.

  • Possible outcomes include:

  • RR (red on 1st, red on 2nd)

  • RB (red on 1st, blue on 2nd)

  • BR (blue on 1st, red on 2nd)

  • BB (blue on 1st, blue on 2nd)

Permutations

Definition

• Permutation: An arrangement of objects where the order is significant.

• Factorial Notation:

  • $n!$ (factorial of n) is equal to the product of all positive integers up to n.

  • Example: $3! = 3 imes 2 imes 1 = 6$

  • $10! = 10 imes 9 imes 8 imes … imes 1 = 3,628,800$

General Principle

• In general, $n$ distinct objects can be arranged in $n!$ ways.

• For example, with 3 distinct objects, the arrangements are 3! = 6.

Permutations Example: Seating Arrangement

• Example: Compute the number of possible ordered seating arrangements for 8 people in 5 chairs.

• Solution: Using permutations:

  • We are arranging $r=5$ of $n=8$, so use:

  • $P_{8,5} = rac{8!}{(8-5)!} = rac{8!}{3!}$

  • Calculation will show 6720 arrangements.

Combinations

Definition

• Combination: A selection of objects where order does not matter.

Example Comparison

• Choosing students for a graduation ceremony:

  • Choosing which students to give speeches (combination, order irrelevant).

  • Assigning speaking order (permutation, order relevant).

Counting Rule for Combinations

• The formula for the number of combinations of $n$ objects taken $r$ at a time is:

  • $C_{n,r} = rac{n!}{r!(n-r)!}$

  • Where $n$ and $r$ are positive integers and $n \geq r$.

Example 4: Reading Books

• Example: From 10 books, how many groups of 4 can be formed?

• Solution: Since order does not matter:

  • Using combinations: $C_{10,4} = rac{10!}{4!(10-4)!}$ = 210 groups

Properties of Combinations

• General properties:

  • $C_{n,1} = n$

  • $C_{n,n} = 1$

  • $C<em>{n,r} = C</em>{n,n-r}$

Summary of Counting Techniques

• Multiplication Rule: Tells the total number of outcomes for a sequence of events.

• Tree Diagrams: Offer a visual representation of outcomes.

• Permutation Rule: For arranging distinct objects where order matters.

• Combination Rule: For forming groups from distinct objects where order does not matter.