Where Did Probability Come From?
People have been playing games of chance for thousands of years. Ancient dice made from animal bones have been found in archaeological sites around the world. But for a long time, nobody had a mathematical way to figure out which outcomes were more or less likely.
It took a couple of curious minds to turn luck into a real branch of mathematics. The idea of probability (a number that measures how likely something is to happen) grew out of questions about card games and dice. Let's see how it developed!
On the ISEE, you won't just need to find one probability. You'll need to compare two or more probabilities and decide which event is more likely, less likely, or equally likely. That's the skill we'll build in this lesson.
Core Principles of Probability
Before you can compare probabilities, you need a solid understanding of what probability means and how it works. Here are the key ideas you'll use again and again.
Probability Is a Number from 0 to 1
The Basic Formula
Compare Using Common Forms
Higher Number = More Likely
Seeing Probability on a Number Line
One of the best ways to compare probabilities is to place them on a probability number line. This line goes from 0 (impossible) to 1 (certain). The diagram below shows several events placed on the line so you can instantly see which is most likely.
The number line makes comparisons easy. You just look at the positions. On the ISEE, you might not draw a number line, but you can picture one in your head. Convert each probability to a decimal or find common denominators, and then see which value lands farther to the right.
The Math Behind Comparing Probabilities
Let's lock in the formulas and techniques you'll use on test day. The starting point is always the same basic formula.
Once you have two probabilities written as fractions, you need a way to compare them. Here are the two main strategies.
Comparing Different Forms of Probability
Sometimes the ISEE will give you probabilities in different forms. One might be a fraction and another might be a percent. You'll need to convert them to the same form before comparing. The diagram below shows how the three forms — fractions, decimals, and percents — connect to each other.
| Given Form | How to Convert | Example |
|---|---|---|
| Fraction → Decimal | Divide numerator by denominator | 3/8 → 3 ÷ 8 = 0.375 |
| Fraction → Percent | Convert to decimal, then multiply by 100 | 3/8 → 0.375 → 37.5% |
| Percent → Decimal | Divide by 100 (move decimal point left 2 places) | 45% → 0.45 |
| Decimal → Fraction | Write as fraction over power of 10, then simplify | 0.6 → 6/10 → 3/5 |
Worked Example: Which Event Is More Likely?
Let's walk through a full problem, step by step, exactly the way you would on the ISEE.
Strategies and Common Pitfalls
On the ISEE, comparing probabilities can be quick — if you use the right strategy. But there are also traps that can slow you down. Here's a guide to the best methods and the mistakes to avoid.
| Strategy ✅ | Common Pitfall ❌ |
|---|---|
| Convert all probabilities to the same form (decimals work great) before comparing. | Comparing fractions with different denominators without converting — for example, thinking 3/8 > 2/5 because 3 > 2. |
| Use cross-multiplication to quickly compare two fractions: a/b vs c/d → compare a × d with c × b. | Crossing the wrong numbers — always multiply the numerator of one fraction by the denominator of the other. |
| Double-check that your total outcomes include everything (all sides of a die, all cards in a deck, etc.). | Forgetting to count all items. A standard die has 6 sides and a deck has 52 cards. Don't assume 50! |
| Estimate when exact calculation is hard. If one fraction is clearly bigger than 1/2 and the other is clearly smaller, you know the answer. | Spending too long on exact division. Sometimes a quick estimate is all you need. |
Connection to Future Math
The skill of comparing probabilities is a stepping stone to more advanced topics you'll see in later math classes. Here's a quick peek at what's ahead.
| What You Know Now | What Comes Next |
|---|---|
| Compare P(A) vs P(B) using fractions and decimals | Compare P(A and B) vs P(A or B) — compound probability |
| Find probability of a single event | Find probability of two events happening in a row (independent events) |
| Use a number line from 0 to 1 | Use probability distributions and tree diagrams |
| Compare theoretical probabilities | Compare theoretical probability with experimental results from real data |
For now, the ISEE only tests single-event probability — so you don't need to worry about compound probability or tree diagrams. Focus on mastering fraction and decimal comparisons. That foundation will make everything else click later on.
Practice Problems
Time to practice! These five problems go from easier to harder. Remember: on the ISEE, always pick an answer for every question — there is no penalty for guessing.
Lesson Summary
To compare probabilities, first calculate each probability using favorable outcomes ÷ total outcomes. Then convert both probabilities to the same form — common denominators, decimals, or percents. The event with the larger number is more likely. If both numbers are equal, the events are equally likely.
Quick ISEE strategies: use cross-multiplication to compare fractions fast, estimate with benchmarks like 1/2 when exact math is slow, and always answer every question since there is no penalty for guessing. Probability values always fall between 0 (impossible) and 1 (certain).