Compute probability of simple events - TACHS Math

SECTION 1

Where Did Probability Come From?

Have you ever wondered how likely it is that your favorite team will win a game? Or what the chances are of rolling a six on a die? People have been asking questions like these for hundreds of years. The study of probability (the math of chance) grew out of games, puzzles, and real-life decisions.

Long ago, people played dice games without really understanding the math behind them. It took some very curious thinkers to figure out the rules of chance. Let's look at a few key moments in the history of probability.

1494 • Luca Pacioli's Puzzle

Italian mathematician Luca Pacioli posed a famous problem about splitting prize money in an unfinished game. This puzzle got mathematicians thinking about how to measure chance.

1654 • Pascal and Fermat's Letters

French mathematicians Blaise Pascal and Pierre de Fermat exchanged letters about gambling problems. Their work laid the foundation for modern probability theory.

1713 • Bernoulli's Big Book

Jacob Bernoulli published a book that showed how probability works over many trials. He proved that the more you repeat an experiment, the closer results get to the predicted probability.

1814 • Laplace's Classical Definition

Pierre-Simon Laplace wrote down the formula we still use today: probability equals favorable outcomes divided by total outcomes. This is the formula you will learn in this lesson!

So here is the big question this lesson answers: If you know all the possible outcomes, how do you calculate the chance that a specific outcome will happen? Let's find out.

SECTION 2

Core Principles & Definitions

Before we start calculating, you need to know a few key vocabulary words. These are the building blocks of every probability problem you will see on the TACHS.

Experiment

An experiment is any action where the result is uncertain. Examples: flipping a coin, rolling a die, or picking a marble from a bag.

Outcome

An outcome is one possible result. For a coin flip, the outcomes are heads and tails.

Sample Space

The sample space is the set of ALL possible outcomes. For a standard die, the sample space is {1, 2, 3, 4, 5, 6}.

Event

An event is one or more outcomes you care about. "Rolling an even number" is an event that includes the outcomes 2, 4, and 6.

Favorable Outcomes

The favorable outcomes are the outcomes that match the event you want. If the event is "rolling a 3," there is 1 favorable outcome.

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KEY TAKEAWAY

Think of probability like a pizza. The whole pizza is the sample space (all possible outcomes). The slices you want are the favorable outcomes. Probability tells you what fraction of the pizza you get!

SECTION 3

Seeing Probability in Action

A picture can make probability much easier to understand. The diagram below shows a bag with 10 marbles of different colors. Imagine you reach in without looking and grab one marble. The diagram highlights which marbles are favorable outcomes for the event "picking a blue marble."

The bag contains 3 blue (B), 2 red (R), 2 green (G), 2 yellow (Y), and 1 purple (P) marble. The three blue marbles are the favorable outcomes. The probability of picking a blue marble is 3 out of 10, or 3/10.

Notice how we counted the marbles we want (blue = 3) and put that over the total number of marbles (10). That fraction, 3/10, is the probability. You can also write it as a decimal (0.3) or a percent (30%). All three forms mean the same thing.

SECTION 4

The Probability Formula

Here is the formula you need to know. It works for any simple event (an event where every outcome is equally likely).

PROBABILITY OF A SIMPLE EVENT

P(event) = Number of favorable outcomes ÷ Total number of outcomes

P(event) = the probability of the event happening. Favorable outcomes = the outcomes that match what you want. Total outcomes = all possible outcomes in the sample space.

Probability always gives you a number between 0 and 1 (or between 0% and 100%). Here is what those boundaries mean.

IMPOSSIBLE EVENT

P(event) = 0

If there are zero favorable outcomes, the event cannot happen. Example: rolling a 7 on a standard die.

CERTAIN EVENT

P(event) = 1

If every outcome is favorable, the event is certain to happen. Example: rolling a number less than 7 on a standard die.

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Quick Tip

If your answer is greater than 1 or less than 0, go back and check your work. Probability can never be negative or more than 1!

SECTION 5

The Probability Scale

It helps to picture probability on a number line from 0 to 1. The closer a probability is to 1, the more likely the event is. The closer it is to 0, the less likely. The diagram below shows where common events fall on this probability scale.

The probability scale runs from 0 (impossible) to 1 (certain). Events near 0.5 are equally likely to happen or not happen. Notice that every probability on this scale is between 0 and 1.

You can express probability as a fraction, a decimal, or a percent. On the TACHS, you may see any of these forms. To convert a fraction to a decimal, divide the top number by the bottom number. To convert a decimal to a percent, multiply by 100.

Event	Fraction	Decimal	Percent
Flipping heads (coin)	1/2	0.5	50%
Rolling a 3 (die)	1/6	≈ 0.167	≈ 16.7%
Drawing a heart (cards)	13/52 = 1/4	0.25	25%

Common probability values in three forms

SECTION 6

Worked Example: Spinner Problem

Let's work through a full problem step by step. Imagine a spinner divided into 8 equal sections. The sections are colored: 3 red, 2 blue, 2 green, and 1 yellow. What is the probability of spinning blue?

Finding P(Blue) on a Spinner

1
Step 1 — Identify the total outcomes

The spinner has 8 equal sections, so there are 8 total possible outcomes.

Total outcomes = 8
2
Step 2 — Count the favorable outcomes

We want to spin blue. There are 2 blue sections on the spinner. These are our favorable outcomes.

Favorable outcomes = 2
3
Step 3 — Plug into the formula

Use the probability formula: P(Blue) = favorable outcomes ÷ total outcomes = 2 ÷ 8.

P(Blue) = 2/8
4
Step 4 — Simplify the fraction

Both 2 and 8 can be divided by 2. So 2/8 simplifies to 1/4.

P(Blue) = 1/4 = 0.25 = 25%

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REMEMBER

Always simplify your fraction if you can! On multiple-choice tests like the TACHS, the answer choices usually show the simplified fraction. If you leave your answer as 2/8 instead of 1/4, you might not find it in the choices.

SECTION 7

Common Mistakes & How to Avoid Them

Probability is straightforward once you get the hang of it. But there are a few traps that trip students up. The table below shows common mistakes and how to fix them.

Common Mistake	Why It's Wrong	How to Fix It
Putting total on top	Writing 8/2 instead of 2/8 flips the fraction and gives a number greater than 1.	Favorable outcomes always go on TOP. Total outcomes always go on the BOTTOM.
Forgetting to count all outcomes	If you miss some outcomes, your total is wrong and the probability will be incorrect.	List every outcome in the sample space before you start. Count carefully!
Not simplifying	The answer 4/12 is correct, but it might not appear in the answer choices.	Always reduce your fraction to lowest terms. Divide top and bottom by their GCF.
Confusing "at least" with "exactly"	"At least 3" means 3, 4, 5, 6 on a die — not just 3.	Read the problem carefully. Underline the key words before counting outcomes.

Watch out for these common probability errors

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KEY TAKEAWAY

Think of probability like making a fraction in math class. The part you want goes on top, and the whole group goes on the bottom. If your answer is more than 1 or less than 0, something went wrong.

SECTION 8

Simple Events vs. Compound Events

In this lesson, you learned about simple events — events with a single outcome or group of outcomes from one experiment. As you move on in math, you will also study compound events (events that combine two or more experiments, like flipping a coin AND rolling a die). Here is a quick comparison so you know what's coming.

Feature	Simple Event	Compound Event
Number of experiments	One (e.g., one die roll)	Two or more (e.g., roll a die AND flip a coin)
Formula	P = favorable ÷ total	May use multiplication or addition rules
Example question	What is P(rolling a 4)?	What is P(rolling a 4 AND flipping heads)?
TACHS focus	Yes — tested often	Sometimes tested at a basic level

How simple events compare to compound events

For now, focus on mastering simple events. Once you can confidently use the P = favorable ÷ total formula, compound events will feel much easier when you learn them later.

SECTION 9

Practice Problems

Try these five problems on your own. They start easy and get harder. After each question, check the answer and read the explanation.

Question 1: CONCEPTUAL

A jar has 5 red gumballs and 5 blue gumballs. You pick one without looking. Is the probability of picking red greater than, less than, or equal to 1/2? Explain your reasoning.

Show answer

The probability of picking red is equal to 1/2. There are 5 favorable outcomes (red gumballs) out of 10 total gumballs. P(red) = 5/10 = 1/2. Since half the gumballs are red, you have an equal chance of picking red or blue.

Question 2: BASIC CALCULATION

A standard die has faces numbered 1 through 6. What is the probability of rolling a number greater than 4? Write your answer as a simplified fraction.

Show answer

The numbers greater than 4 on a die are 5 and 6. That gives us 2 favorable outcomes out of 6 total outcomes. P(greater than 4) = 2/6 = 1/3. We simplified by dividing both the numerator and the denominator by 2. As a decimal, this is about 0.333, or roughly 33.3%.

Question 3: INTERMEDIATE

A bag contains 4 green marbles, 3 red marbles, 2 yellow marbles, and 1 white marble. What is the probability of NOT picking a green marble?

Show answer

First, find the total: 4 + 3 + 2 + 1 = 10 marbles. The event is "NOT green," so the favorable outcomes are all the non-green marbles: 3 + 2 + 1 = 6. P(not green) = 6/10 = 3/5. Another way to solve this: P(not green) = 1 − P(green) = 1 − 4/10 = 6/10 = 3/5. Both methods give the same answer.

Question 4: APPLIED

A class has 12 boys and 18 girls. The teacher puts every student's name in a hat and draws one name for a prize. What is the probability that the winner is a girl? Express your answer as a fraction, a decimal, and a percent.

Show answer

Total students = 12 + 18 = 30. Favorable outcomes (girls) = 18. P(girl) = 18/30. Simplify by dividing top and bottom by 6: 18/30 = 3/5. As a decimal: 3 ÷ 5 = 0.6. As a percent: 0.6 × 100 = 60%. So there is a 60% chance the winner is a girl. This makes sense because girls make up more than half the class.

Question 5: CRITICAL THINKING

Maria says, "I flipped a coin 10 times and got heads 7 times, so the probability of heads must be 7/10." Is Maria correct? Explain why or why not, and state what the theoretical probability of heads actually is.

Show answer

Maria is not correct. She found the experimental (observed) probability, not the theoretical probability. The theoretical probability of heads on a fair coin is always 1/2 because there is 1 favorable outcome (heads) out of 2 possible outcomes (heads or tails). In a small number of trials, results can differ from the theoretical probability. If Maria flipped the coin hundreds of times, her results would likely get closer and closer to 50%. This idea is related to the Law of Large Numbers.

SUMMARY

Lesson Summary

Summary

Probability measures how likely an event is to happen. For a simple event where all outcomes are equally likely, use the formula: P(event) = favorable outcomes ÷ total outcomes. The answer is always a number from 0 (impossible) to 1 (certain). You can write probability as a fraction, decimal, or percent.

To solve a probability problem: first identify the sample space (all possible outcomes), then count the favorable outcomes (the ones that match the event you want), and finally divide. Always simplify your fraction so your answer matches the choices on the TACHS. Remember: favorable on top, total on the bottom!