Compute probability in simple situations (as included in math content) - TACHS Math

SECTION 1

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.

1564 • Cardano's Book of Games

Italian mathematician Gerolamo Cardano wrote the first known book about probability. He studied dice games and figured out how to count outcomes.

1654 • Pascal and Fermat

French mathematicians Blaise Pascal and Pierre de Fermat exchanged letters about gambling problems. Their work created the foundations of modern probability.

1713 • Bernoulli's Big Idea

Jacob Bernoulli published a book showing that the more times you repeat an experiment, the closer your results get to the true probability.

Today • Probability Everywhere

Probability is now used in weather forecasting, medicine, sports analytics, video games, and much more.

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!

SECTION 2

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

An experiment is any action where the result is uncertain. Flipping a coin, rolling a die, or picking a marble from a bag are all experiments.

Outcome

An outcome is one possible result. For example, when you flip a coin, "heads" is one outcome and "tails" is another.

Sample Space

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

Event

An event is the specific outcome (or group of outcomes) you care about. "Rolling an even number" is an event that includes 2, 4, and 6.

Favorable Outcomes

These are the outcomes that match the event you want. If your event is "rolling a 5," then there is only 1 favorable outcome.

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KEY TAKEAWAY

Think of probability like picking a slice of pizza from a whole pie. The whole pie is your sample space (all the slices). The slices you actually want (say, pepperoni) are your favorable outcomes. Probability tells you what fraction of the pie is pepperoni!

SECTION 3

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.

The probability number line ranges from 0 (impossible) to 1 (certain). Notice how everyday events, like flipping a coin, land right in the middle at 1/2.

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.

SECTION 4

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.

BASIC PROBABILITY FORMULA

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

P(event) means "the probability of the event happening." Favorable outcomes are the results you want. Total outcomes is the full sample space.

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?

MARBLE EXAMPLE

P(red) = 3 ÷ 10 = 3/10 = 0.3 = 30%

There are 3 favorable outcomes (red marbles) out of 10 total outcomes (all marbles).

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.

COMPLEMENT RULE

P(NOT event) = 1 − P(event)

If the probability of picking red is 3/10, then the probability of NOT picking red is 1 − 3/10 = 7/10.

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TACHS Tip

Always check that your probability answer is between 0 and 1 (or between 0% and 100%). If you get a number bigger than 1, go back and check your work!

SECTION 5

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.

Four common probability setups: coin flips (2 outcomes), dice (6 outcomes), marble bags (varies), and spinners (varies). Each experiment has equally likely outcomes, so you can use the basic formula.

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%

Quick reference for common probability experiments

SECTION 6

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.

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Problem

You randomly draw one card from a standard 52-card deck. What is the probability that you draw a heart?

Finding P(heart)

1
Step 1 — Identify the total outcomes

A standard deck has 52 cards. Since you draw one card at random, every card has an equal chance of being picked. So the total number of outcomes is 52.

Total outcomes = 52
2
Step 2 — Count the favorable outcomes

The event we want is "drawing a heart." There are 13 hearts in a deck (Ace through King of hearts). So there are 13 favorable outcomes.

Favorable outcomes = 13
3
Step 3 — Plug into the formula

Use the probability formula: P(heart) = favorable ÷ total = 13 ÷ 52.

P(heart) = 13/52
4
Step 4 — Simplify the fraction

Both 13 and 52 are divisible by 13. So 13 ÷ 13 = 1 and 52 ÷ 13 = 4. The simplified fraction is 1/4.

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

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%.

SECTION 7

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).

Avoid these common mistakes on TACHS probability questions

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KEY TAKEAWAY

Think of the probability formula like a basketball free-throw stat. If a player made 7 out of 10 free throws, their success rate is 7/10. The shots made (favorable) go on top. The total shots attempted go on the bottom. Probability works the same way!

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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

How simple probability connects to future math

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.

SECTION 9

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.

Question 1: CONCEPTUAL

A jar has 5 green gumballs and 5 yellow gumballs. Without looking, you reach in and grab one. Is the probability of picking a green gumball closer to 0, 1/2, or 1?

Show answer

The probability is exactly 1/2. There are 5 green gumballs out of 10 total, so P(green) = 5/10 = 1/2. Since half the gumballs are green and half are yellow, you have an equal chance of picking either color. This is right in the middle of the probability number line.

Question 2: BASIC CALCULATION

You roll a standard six-sided die. What is the probability of rolling a number less than 3? Express your answer as a fraction in simplest form.

Show answer

The numbers less than 3 on a die are 1 and 2. That gives us 2 favorable outcomes. The total number of outcomes is 6. Using the formula: P(less than 3) = 2/6. Simplify by dividing both the numerator and denominator by 2. The answer is 1/3.

Question 3: INTERMEDIATE

A bag contains 4 red marbles, 6 blue marbles, and 2 white marbles. You pick one marble at random. What is the probability that you do NOT pick a blue marble? Give your answer as a fraction, a decimal, and a percent.

Show answer

First find the total: 4 + 6 + 2 = 12 marbles. The probability of picking blue is P(blue) = 6/12 = 1/2. Now use the complement rule: P(NOT blue) = 1 − 1/2 = 1/2. Alternatively, the non-blue marbles are 4 red + 2 white = 6, so P(NOT blue) = 6/12 = 1/2. The answer is 1/2 = 0.5 = 50%.

Question 4: APPLIED

A teacher puts the names of 30 students into a hat. There are 12 boys and 18 girls. She draws one name at random to win a prize. What is the probability that the winner is a girl? A second teacher says, "There is a 50-50 chance it's a girl." Is that teacher correct? Explain.

Show answer

The total number of names is 30. The favorable outcomes (girl names) = 18. So P(girl) = 18/30. Simplify by dividing both by 6: 18 ÷ 6 = 3 and 30 ÷ 6 = 5. So P(girl) = 3/5 = 0.6 = 60%. The second teacher is incorrect. A 50-50 chance would mean P = 1/2 = 50%, but the actual probability is 60%. Since there are more girls than boys, a girl is more likely to be chosen.

Question 5: CRITICAL THINKING

A spinner has 10 equal sections numbered 1 through 10. Maria says the probability of landing on a prime number is 1/2. Is she right? (Hint: Remember that 1 is NOT a prime number.) List the prime numbers from 1 to 10 to support your answer.

Show answer

A prime number has exactly two factors: 1 and itself. The numbers from 1 to 10 are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. The number 1 is NOT prime. The primes are 2, 3, 5, and 7. That is 4 prime numbers out of 10 total. So P(prime) = 4/10 = 2/5 = 0.4 = 40%. Maria is incorrect. She probably counted 1 as a prime, which would have given her 5/10 = 1/2. Since 1 is not prime, the correct probability is only 2/5.

SUMMARY

Probability — Quick Review

Summary

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!