Where Did Probability Come From?
Have you ever wondered why some things happen more often than others? People have asked that question for hundreds of years. Probability is the area of math that helps us measure how likely something is to happen. It all started with games of chance — dice, cards, and coin flips.
Long ago, gamblers noticed patterns when they rolled dice. They wanted to know: "If I roll two dice, what are my chances of getting a certain number?" That simple question led to one of the most useful branches of mathematics.
So here is the big question probability answers: Out of all the things that could happen, how many of them are the result I want? Let's learn how to figure that out!
Core Principles of Probability
Before we start calculating, you need to know a few key words. These ideas are the building blocks of every probability problem you will see on the TACHS.
Experiment
Outcome
Sample Space
Event
Favorable Outcomes
Seeing Probability on a Number Line
Probability is always a number from 0 to 1. A probability of 0 means the event is impossible. A probability of 1 means the event is certain. Most events fall somewhere in between. The diagram below shows where common probabilities land on a number line.
You can express probability as a fraction (like 1/2), a decimal (like 0.5), or a percent (like 50%). They all mean the same thing. On the TACHS, you might see the answer choices in any of these forms. Make sure you are comfortable converting between them.
The Probability Formula
Here is the formula you will use for almost every simple probability problem. It is straightforward once you know what each part means.
Let's say you have a bag with 3 red marbles and 7 blue marbles. You reach in and grab one marble without looking. What is the probability of picking a red marble?
There is another handy formula. The probability that an event does NOT happen is called the complement. You can find it by subtracting from 1.
Common Types of Probability Situations
On the TACHS, you will see a few common setups. Knowing these ahead of time will save you time on test day. Let's look at the most popular ones.
| Experiment | Total Outcomes | Example Event | P(Event) |
|---|---|---|---|
| Coin flip | 2 | Heads | 1/2 = 0.5 = 50% |
| Standard die | 6 | Rolling a number > 4 | 2/6 = 1/3 ≈ 33.3% |
| Bag of 10 marbles | 10 | Picking a green marble (4 green) | 4/10 = 2/5 = 40% |
| Spinner with 8 sections | 8 | Landing on an odd number | 4/8 = 1/2 = 50% |
Worked Example: Picking a Card
Let's walk through a full problem step by step, just like you would on the TACHS. A standard deck of playing cards has 52 cards. There are 4 suits (hearts, diamonds, clubs, spades), each with 13 cards. Hearts and diamonds are red. Clubs and spades are black.
There is a 25% chance of drawing a heart. That makes sense because hearts are one of four equally sized suits, and 1 out of 4 = 25%.
Common Mistakes vs. Best Practices
Many students make the same small errors on probability problems. Knowing these mistakes ahead of time can help you avoid them on the TACHS.
| Common Mistake | Why It's Wrong | Best Practice |
|---|---|---|
| Forgetting to count ALL outcomes | If you miss some items, your denominator is too small and your answer is too big. | List or add up every item before you divide. |
| Not simplifying the fraction | The correct answer on a multiple-choice test might be in simplest form. | Always reduce your fraction to lowest terms. |
| Writing probability greater than 1 | Probability can never be more than 1 (or 100%). | Check: is your answer between 0 and 1? |
| Confusing "favorable" with "total" | Putting the total on top and the favorable on the bottom flips your answer. | Favorable goes on top (numerator). Total goes on the bottom (denominator). |
From Simple to Compound Probability
On the TACHS, you mainly need simple probability — one experiment, one event. But it helps to see how this connects to bigger ideas you will study later in math.
| Feature | Simple Probability (TACHS Level) | Compound Probability (Future Topic) |
|---|---|---|
| Number of events | One event at a time | Two or more events together |
| Example | Roll a die, get a 5 | Roll a die AND flip a coin |
| Formula | P = favorable ÷ total | May multiply or add probabilities |
| Difficulty | Count and divide | Requires understanding independent and dependent events |
For now, focus on mastering the basics. If you can quickly count favorable outcomes and total outcomes, you have the skill you need. Everything else in probability builds on that foundation.
Practice Problems
Try these five problems on your own. They go from easier to harder. After you work each one, check the answer and explanation.
Probability — Quick Review
Probability measures how likely an event is to happen, and it always falls between 0 (impossible) and 1 (certain). To find it, use the formula: P(event) = favorable outcomes ÷ total outcomes. Always identify the sample space (all possible results) and count the favorable outcomes (the results you want) before plugging into the formula.
Remember to simplify your fraction and make sure your answer is between 0 and 1. You can express probability as a fraction, a decimal, or a percent. Use the complement rule (P(NOT event) = 1 − P(event)) when a problem asks about something NOT happening. With these tools, you are ready to tackle any simple probability question on the TACHS!