Solving Problems Using Outcomes of Experiments

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 lesson, the learners are able to gather data from experiments and represent the data in different forms.

Learning Competencies

1. Express outcomes in words and/or symbols and represent outcomes in tables and/or graphs.

2. Solves problems using the outcomes of experiments.

Learning Objectives

Accurately identify all the possible outcomes in an experiment using systematic listing.

Correctly solve problems involving outcomes in an experiment using systematic listing.

Accurately solve problems using the outcomes of experiments.

Correctly illustrate the probability of simple events.

Accurately solve problems involving the probability of simple events.

Definition of Probability

Mathematical Measure

Probability is the measure of how likely an event is to occur.

Calculation Formula

It is calculated using the formula: Probability = Number of favorable outcomes / Total number of possible outcomes

Value Range

Probability values range from 0 (impossible event) to 1 (certain event)

Outcomes and Sample Space

Outcome

A possible result of an experiment.

Sample Space

The set of all possible outcomes.

Example: If rolling a die, the sample space is {1, 2, 3, 4, 5, 6}.

Types of Probability Events

Certain Event (P = 1)

An event that will always happen.

Impossible Event (P = 0)

An event that will never happen.

Equally Likely Events

Events that have the same probability of occurring.

Mutually Exclusive Events

Events that cannot happen at the same time.

Listing Outcomes Systematically

Enumeration (Listing Method)

Writing all possible outcomes in an organized list.

Tree Diagram

A branching method that shows all outcomes visually.

Table/Grid Method

Organizing outcomes in a table for clarity.

Real-Life Applications of Probability

Games of Chance

Dice, cards, lottery, and other games where probability determines outcomes.

Weather Forecasting

Rain probability and other weather predictions based on statistical models.

Sports Predictions

Analyzing winning chances and player performance using probability models.

Decision-Making

Using probability concepts to make informed choices in everyday life.

Lesson Purpose

Develop Critical Thinking

Apply logical reasoning to predict outcomes and make informed decisions based on probability concepts. Students learn to analyze situations systematically.

Illustrate Probability

Represent probability values in multiple forms including fractions, decimals, and visual models to better understand the likelihood of events occurring.

Solve Probability Problems

Apply systematic methods like listing, tree diagrams, and tables to determine all possible outcomes of an experiment and calculate probabilities accurately.

This lesson aims to help students accurately solve probability problems by applying systematic methods to determine all possible outcomes of an experiment. Students will learn to correctly illustrate the probability of simple events using various representations and develop critical thinking skills to recognize patterns and make informed decisions.

Key Vocabulary

1

Outcome Table

A table used to organize and display all possible outcomes of an experiment.

2

Event Space

A subset of the sample space that contains only the favorable outcomes for a given event.

3

Complementary Events

Two events that together cover all possible outcomes. Example: If event A is rolling an even number, the complement of A is rolling an odd number.

More Key Vocabulary

Probability Model

A mathematical representation of a random experiment, including the sample space and probabilities of events.

Odds

A ratio comparing the number of favorable outcomes to the number of unfavorable outcomes. Example: If there are 3 red and 2 blue marbles, the odds of drawing red are 3:2.

Uniform Probability Model

A model where all outcomes in the sample space are equally likely to occur.

Non-Uniform Probability Model

A probability model where outcomes are not equally likely. Example: A weighted die where some numbers have a higher chance of appearing.

Advanced Probability Concepts

1

Law of Large Numbers

A principle stating that as an experiment is repeated more times, the experimental probability approaches the theoretical probability.

2

Random Variable

A numerical value that represents the outcome of an experiment. Example: The number of heads in 5 coin flips.

3

Fair Game

A game in which all players have an equal chance of winning based on probability.

4

Biased Experiment

An experiment where certain outcomes are more likely due to external factors. Example: A loaded die that rolls a 6 more frequently.

Combinatorial Concepts

Permutation

An arrangement of objects in a specific order. Example: The number of ways to arrange three letters (ABC, ACB, BAC, etc.).

Combination

A selection of objects where the order does not matter. Example: The number of ways to choose two letters from A, B, and C (AB, AC, BC).

Statistical Probability Concepts

Expected Value

The predicted average outcome of a probability experiment over the long run. Example: The expected number of heads in 100 coin flips is 50.

Frequency Distribution

A table or graph showing how often different outcomes occur in an experiment.

Cumulative Probability

The probability of an event occurring up to a certain point. Example: The probability of rolling a number ≤ 4 on a die (1, 2, 3, or 4).

Monte Carlo Simulation

A method that uses repeated random sampling to estimate probability outcomes.

Advanced Probability Theory

1

Conditional Probability

The probability of an event occurring given that another event has already occurred. Example: The probability of drawing an ace from a deck after one ace has already been drawn.

2

Bayes' Theorem

A formula used to find the probability of an event based on prior knowledge of related events.

3

Gambler's Fallacy

The mistaken belief that past events affect the likelihood of future independent events. Example: Thinking that after flipping five heads in a row, the next flip is "due" to be tails.

Solving Problems Involving Probability of Simple Events

A standard deck has 52 cards, including 4 suits (hearts, diamonds, clubs, spades) with 13 cards per suit.

Drawing a Heart Card

Probability of drawing a heart: 13/52 = 1/4

Favorable outcomes: 13 hearts

Total outcomes: 52 cards

Drawing a Face Card

Probability of drawing a face card: 12/52 = 3/13

Favorable outcomes: 12 face cards (3 per suit × 4 suits)

Total outcomes: 52 cards

Rolling a 4 on a Die

Probability of rolling a 4: 1/6

Favorable outcomes: 1 (the face showing 4)

Total outcomes: 6 (all possible faces)

These examples demonstrate how to calculate probability by finding the ratio of favorable outcomes to total possible outcomes in simple events.