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  1. ISEE Lower Level Quantitative Reasoning

  3. Calculate probability from counts or ratios.

ISEE LOWER LEVEL • QUANTITATIVE REASONING

Calculate probability from counts or ratios.

Learn how to figure out the chance of picking something from a group.

SECTION 1

Where Did Probability Come From?

Have you ever wondered what the chances are of something happening? Maybe you wondered if it would rain on your birthday, or if you'd pick the red candy from a bag. People have been asking questions like this for hundreds of years!

Probability is the math word for "chance." It tells us how likely something is to happen. Let's see how people started figuring this out.

1400s
Games of Chance
People played dice games and card games. They started wondering: "Is this game fair?" That curiosity led to the study of probability.
1654
Two Math Friends Write Letters
Two French mathematicians, Blaise Pascal and Pierre de Fermat, wrote letters to each other about how to solve dice problems. This is often called the start of probability math!
1700s
Probability Grows Up
Scientists began using probability to study weather, medicine, and more. It wasn't just about games anymore!
Today
Probability Is Everywhere
We use probability every day — weather forecasts, sports predictions, and yes, even on the ISEE test!

So the big question is: how do we use numbers to figure out the chance of something happening? That's exactly what you'll learn in this lesson. Let's go!

SECTION 2

What Is Probability? The Big Ideas

Probability is all about comparing two numbers. You compare what you want to happen to everything that could happen. Here are the main ideas you need to know.

1

Favorable Outcomes

These are the results you WANT. If you want to pick a red marble, the red marbles are your favorable outcomes.
2

Total Outcomes

This is EVERYTHING that could happen. It's the total number of items in the group — all the marbles in the bag.
3

The Probability Fraction

Probability is a fraction! The number you want goes on top. The total goes on the bottom. That's it!
4

Probability Range

Probability goes from 0 (impossible — no chance!) to 1 (certain — it will definitely happen). Most answers are a fraction between 0 and 1.
✦ KEY TAKEAWAY
Think of probability like a pizza. Imagine a pizza cut into 8 slices. If 3 slices have pepperoni, the probability of grabbing a pepperoni slice is 3 out of 8. The part you want goes on top, and the whole group goes on the bottom. That's all probability is!
SECTION 3

See It! Marbles in a Bag

Let's look at a picture that shows how probability works. Imagine you have a bag with 10 marbles. Some are red, some are blue, and some are green. Take a look!

Marbles in a Bag — 10 TotalTHE BAGRed = 4 marblesBlue = 3 marblesGreen = 3 marblesTotal = 4 + 3 + 3 = 10
This bag has 4 red, 3 blue, and 3 green marbles — 10 total. To find the probability of picking red, put 4 on top and 10 on the bottom: 4/10.

See how easy that is? If you reach in without looking, the probability of picking a red marble is 4 out of 10. We write that as the fraction 4/10, which simplifies to 2/5. The probability of picking a blue marble is 3 out of 10, or 3/10.

SECTION 4

The Probability Formula

Here is the one formula you need for probability questions on the ISEE. It's simple — just a fraction!

PROBABILITY FORMULA
Probability = Number of favorable outcomes ÷ Total number of outcomes
Favorable outcomes = the ones you WANT. Total outcomes = ALL of them added together.

Let's break that down with a simple example. Say there are 5 apples in a basket: 2 are red and 3 are green.

EXAMPLE — RED APPLE
Probability of red = 2 ÷ 5 = 2/5
There are 2 red apples (what you want) out of 5 total apples.
EXAMPLE — GREEN APPLE
Probability of green = 3 ÷ 5 = 3/5
There are 3 green apples (what you want) out of 5 total apples.
💡 ISEE Test Tip!
On the ISEE, always add up ALL the items first to get your total. Sometimes the problem gives you counts of each type, and you need to add them together. Don't forget this step!
SECTION 5

Different Ways Problems Give You Info

ISEE problems might give you the information in different ways. Sometimes you'll get a list of counts. Other times you'll see a table, a picture, or a ratio. Let's look at the main types.

Three Ways You Might See Probability InfoFROM COUNTS"A jar has 5 red,3 blue, and 2 yellowgumballs."Step 1: Add them up5 + 3 + 2 = 10Step 2: Make fractionP(red) = 5/10Step 3: Simplify= 1/2FROM A TABLEColorCountRed6Blue4Green2Total12Read the table!P(blue) = 4/12= 1/3FROM A RATIO"The ratio of cats todogs is 3 to 2."Step 1: Add ratio parts3 + 2 = 5 partsStep 2: Make fractionP(cat) = 3/5P(dog) = 2/5⚠ The total is the SUMof the ratio numbers!
Three common ways ISEE problems present probability information: as word counts, in a table, or as a ratio. In every case, you build a fraction with what you want on top and the total on the bottom.

The left panel shows a word problem with counts — you just add them up for the total. The middle panel shows a table — read the numbers and find the total row. The right panel shows a ratio — and here's the tricky part: you add the ratio numbers together to get the total. Watch out for that on the test!

SECTION 6

Let's Solve One Together!

Here's a problem just like one you might see on the ISEE. Let's walk through it step by step.

🎯 Practice Problem
A spinner has 8 equal sections: 3 are blue, 2 are red, 1 is yellow, and 2 are green. If you spin the spinner once, what is the probability that it lands on blue?

Solving the Spinner Problem

Step 1 — Find the Total

Add up all the sections: 3 blue + 2 red + 1 yellow + 2 green = 8 total sections. The problem already tells us there are 8 sections, but it's smart to check by adding!
Total outcomes = 8

Step 2 — Find the Favorable Outcomes

We want blue. How many blue sections are there? The problem says 3. So the favorable outcomes = 3.
Favorable outcomes = 3

Step 3 — Build the Fraction

Put the favorable outcomes on top and the total on the bottom.
Probability = 3/8

Step 4 — Simplify If You Can

Can 3/8 be simplified? The only factors of 3 are 1 and 3. The only factors of 8 are 1, 2, 4, and 8. They don't share any common factor besides 1, so 3/8 is already in simplest form.
Answer: 3/8
⭐ ISEE Strategy
Always simplify your fraction and then look at the answer choices. If your answer matches one of the choices, you're done! If not, double-check your counting. On the ISEE, there is no penalty for guessing, so always pick an answer — even if you're not sure!
SECTION 7

Common Mistakes and Smart Tips

Even smart students make small mistakes on probability questions. Let's look at the most common traps and how to avoid them!

Watch out for these traps on test day!
Common MistakeWhy It's WrongWhat to Do Instead
Forgetting to add up the totalYou use a part as the total, making your fraction wrongAlways add ALL groups together first
Putting the total on topThe fraction is upside down! 10/3 instead of 3/10What you WANT goes on TOP
Not simplifying the fractionYour answer might not match any choiceAlways check if you can reduce
Using a ratio number as the totalIn a 3:2 ratio, the total is 5, not 3 or 2Add ratio parts together for total
🧠 REMEMBER THIS!
Think of probability like a team picking jerseys. If your soccer team has 12 players and 4 choose blue jerseys, the probability of randomly picking a blue-jersey player is 4 out of 12 = 1/3. The part is always smaller than or equal to the whole. If your fraction is bigger than 1, something went wrong!
SECTION 8

Probability Gets Even Cooler

Right now you're learning the basics of probability. But as you grow up, probability gets used for really cool things! Here's a peek at what's ahead.

Your probability skills will keep growing!
What You Learn NowWhat Comes Later
Finding probability from simple countsCalculating probability of two events happening together
Writing probability as a fractionWriting probability as a decimal or percent
Equal-chance outcomes (like fair spinners)Unequal-chance outcomes (like weighted dice)
One pick from a groupMultiple picks and whether items are replaced

For the ISEE Lower Level, you just need the basics: count the ones you want, count the total, and make a fraction. Master that, and you'll be ready! Everything else builds on this same idea.

SECTION 9

Practice Problems

Time to practice! Try each problem on your own before looking at the answer. Remember: count what you want, count the total, and build your fraction. You've got this!

PROBLEM 1 — CONCEPTUAL
A bag has 6 red marbles and 4 blue marbles. If you pick one marble without looking, which color are you MORE likely to pick?
PROBLEM 2 — BASIC CALCULATION
A jar has 3 red candies, 5 green candies, and 2 yellow candies. What is the probability of picking a green candy?
PROBLEM 3 — INTERMEDIATE
A spinner is divided into 12 equal sections. Four sections are labeled "Win" and the rest are labeled "Try Again." What is the probability of landing on "Try Again"?
PROBLEM 4 — APPLIED
At a pet store, the ratio of fish to birds is 5 to 3. If you randomly pick one animal from the store, what is the probability that it is a bird?
PROBLEM 5 — CRITICAL THINKING
A box has some red, blue, and white buttons. The probability of picking a red button is 1/4, and the probability of picking a blue button is 1/2. If there are 20 buttons total, how many white buttons are in the box?
SUMMARY

Let's Wrap It Up!

Probability tells you the chance of something happening. To calculate it, you make a fraction: put the favorable outcomes (what you want) on top and the total outcomes (everything) on the bottom. Always add up all the groups to find your total, and simplify your fraction at the end.

Watch out for ratio problems — remember to add the ratio parts together to get the total. Your probability should always be between 0 and 1. If it's bigger than 1, double-check your work. On the ISEE, never leave a question blank — always pick an answer! You've got this! 🎉

Varsity Tutors • ISEE Lower Level • Calculate probability from counts or ratios.