Experiments, Sample Space, Simple Events, and Outcomes

Content Standards and Learning Competencies

Content Standards

The learners should have knowledge and understanding of outcomes from experiments.

Performance Standards

By the end of the quarter, the learners are able to gather data from experiments and represent the data in different forms.

Learning Competencies

1. Collect data from experiments (e.g., number of heads obtained when tossing a coin, number of times, number of prime numbers obtained when rolling a die).

2. Express outcomes in words and/or symbols and represents outcomes in tables and/or graphs.

Learning Objectives

1

Accurately collect data from experiments

Students will learn proper techniques for gathering data from probability experiments such as coin tosses and dice rolls.

2

Correctly determine and identify the experiment, outcomes, sample space, and events

Students will be able to identify these key components in any given probability situation.

3

Accurately count and list the number of occurrences of an outcome

Students will develop skills to systematically track and record experimental outcomes.

Activating Prior Knowledge: Writing Fractions

Ratio of Cheerleaders to Football Players

There are 32 football players and 16 cheerleaders at your school. Write the ratio of cheerleaders to football players.

Classroom Fractions

Using the table with Classroom A (10 boys, 22 girls) and Classroom B (15 boys, 19 girls), find:

a. fraction of girls to boys in classroom A

b. fraction of boys in classroom A to boys in classroom B

c. fraction of girls to the total number of students in both classes

Flower Fractions

Write the fraction in simplest form:

a. sunflower to rose

b. rose to the total number of flowers

Key Probability Terms

Terms

Definition

Experiments

Activities that produce an outcome, but their results cannot be precisely predicted.

Sample Space

The set of all the possible outcomes of an experiment.

Event

One or more of the outcomes of an experiment.

Outcome

The possible results of an experiment.

What is Sample Space?

The sample space (denoted as S) is the set of all possible outcomes in a probability experiment.

Definition

The sample space (denoted as S) is the set of all possible outcomes in a probability experiment. It is often written using set notation.

Example 1: Flipping a Coin

The only possible results are Heads (H) or Tails (T).

Sample space: S={H,T}

Example 2: Rolling a Six-Sided Die

A die has six faces, numbered 1, 2, 3, 4, 5, and 6.

Sample space: S={1,2,3,4,5,6}

Ways to Represent Sample Space: Listing Method

Listing Method

This method lists all possible outcomes inside curly brackets { }.

Example: Tossing two coins

S={HH,HT,TH,TT}

Example: Rolling a die

S={1,2,3,4,5,6}

Example: Drawing a card from standard deck

S={A♠,2♠,...,K♠,A♥,2♥,...,K♥,A♦,2♦,...,K♦,A♣,2♣,...,K♣}

Ways to Represent Sample Space: Table Method

The table method is used for multiple events, like rolling two dice. It helps us visualize all possible combinations in a structured format.

Table for Two Dice

A 6×6 table showing all 36 possible outcomes when rolling two dice. Each cell represents a specific combination of values.

Table for Two Coins

A 2×2 table showing the four possible outcomes when tossing two coins: HH, HT, TH, and TT.

Card Deck Table

A table organizing a standard deck of 52 cards by suits (rows) and values (columns) to show the complete sample space.

Tables like these help students visualize all possible outcomes in a probability experiment, making it easier to identify specific events and calculate their probabilities.

Ways to Represent Sample Space: Tree Diagram

A tree diagram is useful when dealing with multiple stages. It visually represents how outcomes branch out at each stage of an experiment.

Flipping a Coin Twice

The tree branches out for each flip, showing all possible outcomes: Heads-Heads (HH), Heads-Tails (HT), Tails-Heads (TH), and Tails-Tails (TT).

Drawing Colored Marbles

This tree diagram shows the possible outcomes when drawing marbles of different colors, illustrating how sample space expands with each choice.

Rolling a Die and Flipping a Coin

This diagram shows the combined sample space when performing two different random experiments in sequence.

From these diagrams, we can see how tree diagrams help us visualize and list all possible outcomes in multi-stage probability experiments.

Calculating Probability Using Sample Space

For example, when rolling a six-sided die, the probability of rolling a 4 is:

This is because there is only one 4 in the sample space of six outcomes.

The formula for calculating probability is:

P(Event) = Number of favorable outcomes / Total number of possible outcomes

This demonstrates why having a complete and accurate sample space is essential for probability calculations.

Real-Life Applications of Sample Space

Weather Forecasting

Predicting if it will rain or not based on historical data and current conditions.

Sports

Calculating the chance of winning a game or predicting tournament outcomes.

Business & Finance

Estimating risks in investments and making data-driven business decisions.

Medicine

Determining the probability of side effects from a drug or treatment outcomes.

Key Points About Sample Space

Foundation of Probability

Sample space is the foundation of probability, providing the complete set of possible outcomes.

Decision Making

Understanding sample space helps in making predictions and informed decisions in uncertain situations.

Multiple Representations

It can be represented using lists, tables, and tree diagrams depending on the complexity of the experiment.

Worked Example 1: Rolling a Die

A die is rolled. If it lands on an even number, identify the experiment, outcomes, sample space, and event in this scenario.

Experiment

Rolling a die once is our experiment

Sample Space

Sample Space: S = {1, 2, 3, 4, 5, 6}

All possible outcomes when rolling a die

Event

The event of rolling an even number: {2, 4, 6}

This is the subset of outcomes we're interested in

Solution: When identifying the components of this probability scenario, we separate the experiment (rolling the die), the complete sample space (all six possible outcomes), and the specific event we're interested in (rolling an even number).

Representing Example 1 with Table and Tree Diagram

There are different ways to visualize the sample space when rolling a die. Here are two effective methods:

Table Representation

The table method organizes outcomes in rows and columns, making it easy to identify even numbers (2, 4, 6) among all possible outcomes when rolling a die.

Tree Diagram

The tree diagram shows all possible outcomes as branches, providing a visual way to trace through the sample space and identify the even-numbered outcomes.

Both representations help us visualize that the sample space S = {1, 2, 3, 4, 5, 6} contains three even numbers {2, 4, 6}, giving us P(even) = 3/6 = 1/2.

Worked Example 2: Drawing Marbles

Green, yellow, blue, white, and black marbles are placed in a covered jar. When a marble is randomly drawn from the jar, it turns out to be green.

The Experiment

Drawing a single marble from a jar containing colored marbles

The Sample Space

S = {Green, yellow, blue, white, black}

All possible outcomes when drawing a marble

The Event

Drawing a green marble

This is the specific outcome that occurred

When identifying the components of this probability scenario, we separate the experiment (drawing the marble), the complete sample space (all possible marble colors), and the specific event we're interested in (drawing a green marble).

Worked Example 3: Even Numbers and Numbers Greater Than 2

Write the sample space for the experiment that consists of rolling a single die. Find the events that correspond to the phrases "an even number is rolled" and "a number greater than 2 is rolled."

Sample Space S = {1, 2, 3, 4, 5, 6}

The sample space represents all possible outcomes when rolling a single die.

Event E = {2, 4, 6}

The event "an even number is rolled" consists of the outcomes 2, 4, and 6.

Event T = {3, 4, 5, 6}

The event "a number greater than 2 is rolled" consists of the outcomes 3, 4, 5, and 6.

When working with probability problems, identifying the sample space and defining events as subsets allows us to clearly visualize and calculate probabilities.

Synthesis Activity

Reflect on today's probability concepts by reviewing these key visualizations:

Understanding Sample Space

Sample space represents all possible outcomes of an experiment. Consider what makes a complete sample space when writing your reflection.

Representation Methods

Think about the different ways we represented sample spaces today (listing, tables, tree diagrams) and which you found most helpful.

Applying Probability Concepts

Reflect on how we identify experiments, outcomes, and events in real scenarios, and why understanding sample space matters.

In a one sheet of paper, write what you understand about today's lesson, addressing these key points and any questions you still have.

Evaluation: Multiple Choice Questions

What is the sample space for flipping a fair coin?

a) {H, T}

b) {1, 2, 3, 4, 5, 6}

c) {Even, Odd}

d) {H, HH, HT, T}

How many outcomes are in the sample space when rolling a standard six-sided die?

a) 2

b) 6

c) 12

d) 36

Which of the following represents the correct sample space for tossing two coins?

a) {H, T}

b) {H, HH, T, TT}

c) {HH, HT, TH, TT}

d) {Heads, Tails, Coin}

Evaluation: More Multiple Choice Questions

1

If a die is rolled and a coin is flipped, how many possible outcomes are in the sample space?

a) 6

b) 12

c) 36

d) 24

2

What is the best way to represent the sample space for rolling two dice?

a) List all numbers from 1 to 6

b) Write down all possible sums of the two dice

c) Use a tree diagram or a table to show all 36 outcomes

d) Only write the highest number that appears

Answer Key

Question 1

Answer: A

Sample space for flipping a coin: {H, T}

Question 2

Answer: B

Six outcomes when rolling a standard die

Question 3

Answer: C

Sample space when tossing two coins

Question 4

Answer: B

12 outcomes when rolling a die and flipping a coin

Question 5

Answer: C

Using tables or diagrams to show all 36 outcomes when rolling two dice

Homework Assignment

Read each question carefully

Make sure you understand what is being asked before attempting to solve.

Write down the sample space for each experiment

Be thorough and include all possible outcomes.

Use appropriate representation methods

Use listing, tables, or tree diagrams to represent your answers clearly.

Submit complete and clear solutions

Show all your work and explain your reasoning where appropriate.

Homework Questions

1

Colored Balls

A bag contains three colored balls: red (R), blue (B), and green (G). If one ball is drawn at random, what is the sample space?

2

Coin Flips

A coin is flipped twice. List all possible outcomes.

3

Die Roll

A six-sided die is rolled. What is the sample space?

4

Combined Experiment

A coin is flipped, and a six-sided die is rolled. Create a table to show the sample space.