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SHSAT Math • Probability

Probability of an Event — Determine probability of an event from outcomes.

Learn how to measure the chance of something happening using a simple fraction.

Explore the lesson 10 min read
SHSAT illustration with a school, math function, and reading book
Inside this lesson 10 sections 5 questions 8 figures

SECTION 1

Where Did Probability Come From?

Have you ever wondered why we say there's a "50-50 chance" of getting heads when you flip a coin? People have been asking questions like this for hundreds of years. The math behind chance — called probability — was first developed to solve problems about dice games and card games.

Back in the 1600s, a French gambler asked a mathematician for help winning more games. That question sparked an entire branch of math! Let's look at the key moments that built probability into what you study today.

1564 • Cardano's Book on Games

Italian mathematician Gerolamo Cardano wrote one of the first books about chance. He figured out how to count the ways dice could land.

1654 • Pascal and Fermat Exchange Letters

French mathematicians Blaise Pascal and Pierre de Fermat wrote letters to each other solving a gambling problem. Their work created the foundation of modern probability.

1713 • Bernoulli's Law of Large Numbers

Jacob Bernoulli showed that the more times you repeat an experiment (like flipping a coin), the closer your results get to the predicted probability.

1812 • Laplace's Classic Formula

Pierre-Simon Laplace wrote the probability formula you'll learn today: favorable outcomes divided by total outcomes.

The big question these mathematicians kept asking was: "If I know all the possible results, how can I measure the chance of getting the result I want?" That's exactly what you'll learn in this lesson.

SECTION 2

Core Principles & Definitions

Before you can calculate probability, you need to understand a few key terms. Don't worry — they're simpler than they sound! Think of these as the building blocks for every probability problem you'll see on the SHSAT.

1

Experiment

An experiment is any action where you don't know the result ahead of time. Examples: flipping a coin, rolling a die, or picking a marble from a bag.
2

Outcome

An outcome is one possible result. If you roll a die, one outcome is "3." If you flip a coin, one outcome is "heads."
3

Sample Space

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

Event

An event is the specific outcome (or group of outcomes) you care about. "Rolling an even number" is an event that includes outcomes {2, 4, 6}.
5

Favorable Outcomes

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

KEY TAKEAWAY

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

SECTION 3

Seeing Probability in Action

Let's see what probability looks like with a picture. The diagram below shows a bag with 10 colored marbles. Imagine you reach in without looking and pull out one marble.

Bag of 10 Marbles — What's the Probability of Blue?THE BAG (Sample Space = 10)BBBBRRRGGYCounting Favorable OutcomesBlue = 4Red = 3Green = 2Yellow = 1P(Blue) = 4 / 10 = 2/5
A bag holds 4 blue (B), 3 red (R), 2 green (G), and 1 yellow (Y) marble — 10 total. The probability of picking a blue marble is 4 out of 10, or 2/5.

Notice how the diagram makes counting easy. You can see there are 4 favorable outcomes (blue marbles) and 10 total outcomes (all the marbles). This picture is basically the probability formula in action!

SECTION 4

The Probability Formula

Here is the most important formula for this topic. You'll use it on almost every probability question on the SHSAT. It looks simple, but make sure you really understand what each part means.

BASIC PROBABILITY FORMULA

P(Event)=NumberofFavorableOutcomes÷TotalNumberofOutcomesP(Event) = Number of Favorable Outcomes ÷ Total Number of Outcomes

P(Event) means "the probability that the event happens." The favorable outcomes are the results you want. The total outcomes are all the results that could happen.

PROBABILITY RANGE

0≤P(Event)≤10 ≤ P(Event) ≤ 1

Probability is always between 0 (impossible) and 1 (certain). A probability of 0.5 (or ½) means the event is equally likely to happen or not happen.

COMPLEMENT RULE

P(EventNOThappening)=1−P(Event)P(Event NOT happening) = 1 − P(Event)

If there's a 2/5 chance of picking blue, then there's a 3/5 chance of NOT picking blue. The two probabilities always add up to 1.

💡

SHSAT Tip

On the SHSAT, probability answers are often written as fractions. Always simplify your fraction to lowest terms. For example, write 2/5 instead of 4/10.

SECTION 5

Types of Probability Questions

On the SHSAT, probability questions come in a few common styles. The diagram below shows the most frequent types and how you approach each one. Recognizing the type helps you set up the fraction quickly.

Common Probability Setups on the SHSATDiceSample space: {1,2,3,4,5,6}Total outcomes = 6P(even) = 3/6 = 1/2Favorable: {2, 4, 6}CoinsSample space: {H, T}Total outcomes = 2P(heads) = 1/2Favorable: {H}Cards / MarblesCount each type inthe collectionP(type) = count / totalAdd all counts for totalSpinners / SectionsIf all sections are equal-sized, count sections. If sections are different sizes, use the fraction of the circle.P(section) = number of matching sections / total sections (for equal sections)3-Step Strategy for Any Problem① Count total outcomes② Count favorable outcomes③ Write & simplify
This chart shows the four most common SHSAT probability setups: dice, coins, cards/marbles, and spinners. No matter the setup, the 3-step strategy at the bottom works every time.
SetupTotal OutcomesExample EventFavorableP(Event)
Standard die6Rolling > 42 (5 and 6)2/6 = 1/3
Coin2Getting tails11/2
Bag of 12 marbles12Picking green5 green ones5/12
Spinner with 8 equal sections8Landing on red3 red sections3/8

Quick-reference examples for common probability setups

SECTION 6

Worked Example — Step by Step

Let's walk through a complete SHSAT-style problem together. Follow each step carefully — this is the exact process you'll use on the real test.

ℹ

THE PROBLEM

A jar contains 5 red jelly beans, 3 blue jelly beans, 4 green jelly beans, and 8 yellow jelly beans. If you pick one jelly bean at random, what is the probability that it is blue or green?

Finding P(Blue or Green)

  1. 1

    Step 1 — Find the Total Number of Outcomes

    Add up all the jelly beans in the jar. Red + Blue + Green + Yellow = 5 + 3 + 4 + 8 = 20. So the total number of outcomes is 20.
    Total outcomes = 20
  2. 2

    Step 2 — Count the Favorable Outcomes

    The event is "blue or green." Count the blue jelly beans (3) and the green jelly beans (4). Since the question says "blue or green," you add them together: 3 + 4 = 7.
    Favorable outcomes = 7
  3. 3

    Step 3 — Write the Probability Fraction

    Plug into the formula: P(Blue or Green) = Favorable ÷ Total = 7 ÷ 20.
    P(Blue or Green) = 7/20
  4. 4

    Step 4 — Simplify (if possible)

    Check if 7 and 20 share a common factor. Since 7 is a prime number and does not divide evenly into 20, the fraction is already in simplest form.
    Final Answer = 7/20

SECTION 7

Common Mistakes & How to Avoid Them

Even if you know the formula, small mistakes can cost you points on the SHSAT. Here are the most common errors students make — and how to dodge them.

Common MistakeWhy It's WrongWhat to Do Instead
Forgetting to count ALL items for the totalYou might leave out one color or one category. This makes your denominator too small.Add every group mentioned in the problem. Double-check your sum.
Putting total on top and favorable on bottomThe fraction gets flipped, giving a number greater than 1 (which is impossible for probability).Favorable on TOP, total on BOTTOM. Remember: the part goes over the whole.
Not simplifying the fractionThe answer 4/10 might not appear in the choices; 2/5 will.Always reduce to lowest terms by dividing top and bottom by their GCF.
Confusing "or" with "and""Or" means you ADD the counts. Getting this wrong changes your favorable count."Or" = add the favorable counts. "And" in one draw = look for items that match BOTH conditions.

Top 4 probability mistakes on the SHSAT

✦

KEY TAKEAWAY

Think of probability like a basketball free-throw percentage. If you made 7 out of 20 free throws, your shooting percentage is 7/20. The shots you made go on top, and the total shots go on the bottom. Probability works exactly the same way — what you want goes on top, everything possible goes on the bottom.

SECTION 8

Connecting to Advanced Probability

The formula you learned today is called theoretical probability. It's what you expect should happen based on math. But there's another kind called experimental probability, which is based on what actually happens when you run an experiment. Here's how they compare.

FeatureTheoretical Probability (This Lesson)Experimental Probability (Advanced)
How you find itUse the formula: favorable ÷ totalActually do the experiment and record what happens
ExampleP(heads) = 1/2 because a coin has 2 sidesYou flip a coin 50 times and get heads 23 times, so P(heads) ≈ 23/50
Requires data?No — just countingYes — you need real results
On the SHSAT?Very commonLess common, but possible in data/chart questions

Theoretical vs. Experimental Probability

In high school, you'll learn about compound events (where two things happen, like flipping a coin AND rolling a die), dependent events (where the first result changes the second), and conditional probability. All of these build directly on the simple formula you learned today. Master this, and you'll have a strong head start!

SECTION 9

Practice Problems

Try these five problems on your own. They start easy and get harder. Use the 3-step strategy: count total outcomes, count favorable outcomes, write and simplify the fraction.

Question 1: CONCEPTUAL

A spinner has 4 equal sections colored red, blue, green, and yellow. Is the probability of landing on blue equal to 1/4, 1/2, or 3/4? Explain why.
Show answer
The probability of landing on blue is 1/4. There are 4 equal sections (total outcomes = 4), and only 1 of them is blue (favorable outcomes = 1). Using the formula: P(blue) = 1 ÷ 4 = 1/4. Since all sections are the same size, each color has an equal chance.

Question 2: BASIC CALCULATION

A standard die has faces numbered 1 through 6. What is the probability of rolling a number less than 3?
Show answer
The numbers less than 3 are 1 and 2, so there are 2 favorable outcomes. The total number of outcomes is 6 (since the die has 6 faces). P(less than 3) = 2/6. Simplify by dividing both the numerator and denominator by 2: P(less than 3) = 1/3.

Question 3: INTERMEDIATE

A bag contains 6 red marbles, 4 white marbles, and 2 blue marbles. If you randomly pick one marble, what is the probability of NOT picking a red marble?
Show answer
First, find the total: 6 + 4 + 2 = 12 marbles. You could count the non-red marbles (4 white + 2 blue = 6) and get P(not red) = 6/12 = 1/2. Or you could use the complement rule: P(red) = 6/12 = 1/2, so P(not red) = 1 − 1/2 = 1/2. Both methods give the same answer.

Question 4: APPLIED

A class has 30 students. There are 12 students who play soccer, 8 who play basketball, and 10 who play neither sport. No student plays both. If the teacher randomly picks one student's name from a hat, what is the probability of picking a student who plays a sport?
Show answer
The total outcomes is 30 (all students). Students who play a sport: 12 (soccer) + 8 (basketball) = 20 students play a sport. (You can check: 20 who play + 10 who don't = 30. ✓) P(plays a sport) = 20/30. Simplify by dividing top and bottom by 10: P(plays a sport) = 2/3.

Question 5: CRITICAL THINKING

A number is randomly chosen from the integers 1 through 20. What is the probability that the number is both even AND greater than 14? List the favorable outcomes to support your answer.
Show answer
You need numbers that satisfy BOTH conditions: even AND greater than 14. Numbers greater than 14 from 1–20 are: 15, 16, 17, 18, 19, 20. Out of those, the even ones are 16, 18, and 20. So there are 3 favorable outcomes. The total outcomes are 20 (integers 1 through 20). P(even AND > 14) = 3/20. This fraction is already simplified because 3 and 20 share no common factor other than 1. P(even AND > 14) = 3/20.

SUMMARY

Lesson Summary

Summary

Probability measures the chance of an event happening. You calculate it using the formula: P(Event) = favorable outcomes ÷ total outcomes. The sample space is the full list of possible results, and favorable outcomes are the ones that match the event you want. Probability always falls between 0 (impossible) and 1 (certain).

On the SHSAT, follow the 3-step strategy: (1) count total outcomes, (2) count favorable outcomes, and (3) write the fraction and simplify. Watch out for common traps like flipping the fraction or forgetting to simplify. Use the complement rule (P(not A) = 1 − P(A)) when it's easier to count what you DON'T want. Mastering this formula prepares you for more advanced topics like compound and conditional probability in high school.

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