Compare Probabilities - ISEE Middle Level Mathematics Achievement

SECTION 1

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!

1654 • The Famous Letters

French mathematicians Blaise Pascal and Pierre de Fermat exchanged letters about a gambling problem. Their work became the foundation of probability theory.

1713 • The Law of Large Numbers

Jakob Bernoulli proved that the more times you repeat an experiment, the closer your results get to the true probability. Flip a coin 1,000 times and you'll land near 50% heads.

1814 • Laplace's Big Book

Pierre-Simon Laplace published a major work that organized probability into a clear system. He gave us the classic formula: favorable outcomes divided by total outcomes.

Today • Probability Everywhere

We now use probability in weather forecasts, sports statistics, video game design, medicine, and much more. Comparing probabilities helps us make smart decisions every day.

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.

SECTION 2

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

A probability of 0 means an event is impossible. A probability of 1 means it is certain. Everything else falls in between. You can also express probability as a fraction, decimal, or percent.

The Basic Formula

Probability = favorable outcomes ÷ total outcomes. If a bag has 3 red marbles out of 10 total, P(red) = 3/10.

Compare Using Common Forms

To compare probabilities, convert them to the same form: common denominators, decimals, or percents. Then compare the numbers directly.

Higher Number = More Likely

When probabilities are in the same form, the larger value means a greater chance. If P(A) = 0.6 and P(B) = 0.4, event A is more likely.

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KEY TAKEAWAY

Think of probability like a volume slider on your phone. All the way down (0) means silence — the event will never happen. All the way up (1) means full blast — it's guaranteed. When you compare probabilities, you're checking whose slider is turned up higher.

SECTION 3

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.

Events farther to the right are more likely. Notice that Rolling less than 5 (0.67) is more likely than Flipping Heads (0.50), because its dot sits farther right on the number line.

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.

SECTION 4

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.

BASIC PROBABILITY

P(event) = favorable outcomes ÷ total outcomes

P(event) is the probability of the event happening. Favorable outcomes are the outcomes you want. Total outcomes are all the possible outcomes.

Once you have two probabilities written as fractions, you need a way to compare them. Here are the two main strategies.

STRATEGY 1 — COMMON DENOMINATORS

Rewrite fractions with the same denominator, then compare numerators.

Example: Compare 2/5 and 3/8. The LCD is 40. So 2/5 = 16/40 and 3/8 = 15/40. Since 16 > 15, we know 2/5 > 3/8.

STRATEGY 2 — CONVERT TO DECIMALS

Divide the numerator by the denominator for each fraction, then compare.

Example: 2/5 = 0.4 and 3/8 = 0.375. Since 0.4 > 0.375, we know 2/5 > 3/8.

CROSS-MULTIPLICATION SHORTCUT

To compare a/b and c/d: if a × d > c × b, then a/b > c/d.

Example: Compare 2/5 and 3/8. Compute 2 × 8 = 16 and 3 × 5 = 15. Since 16 > 15, we know 2/5 > 3/8. This is a quick ISEE trick!

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ISEE TIP

On the ISEE, answer choices are ordered from least to greatest. If you can convert each probability to a decimal, you can quickly match it to the right choice. Also, always answer every question — there's no penalty for wrong answers, so take your best guess if you're stuck!

SECTION 5

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.

The top row shows how a single probability (3/4) looks in fraction, decimal, and percent form. The bottom box shows how to compare two probabilities given in different forms: convert both to decimals and compare.

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

Quick reference for converting between probability forms

SECTION 6

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.

A bag holds 5 red marbles, 3 blue marbles, and 2 green marbles. Which is more likely: drawing a red marble or drawing a blue or green marble?

1
Step 1 — Find the Total

Add up all the marbles: 5 + 3 + 2 = 10 total marbles.

Total outcomes = 10
2
Step 2 — Find P(red)

There are 5 red marbles out of 10 total. So P(red) = 5/10 = 1/2.

P(red) = 1/2 = 0.5
3
Step 3 — Find P(blue or green)

There are 3 blue and 2 green marbles, so the favorable outcomes are 3 + 2 = 5. So P(blue or green) = 5/10 = 1/2.

P(blue or green) = 1/2 = 0.5
4
Step 4 — Compare

Both probabilities equal 1/2 or 0.5. The two events are equally likely.

P(red) = P(blue or green) — Equally likely!

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WATCH OUT

Don't forget to add up outcomes for compound events like "blue or green." A common mistake is comparing P(red) with only P(blue) and ignoring P(green). Read the question carefully to see exactly which outcomes count.

SECTION 7

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.

Do's and Don'ts for Comparing Probabilities on the ISEE

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KEY TAKEAWAY

Comparing fractions with different denominators is like comparing two pizzas cut into different numbers of slices. You can't tell who ate more until you cut both pizzas the same way. Common denominators or decimals are how you "cut the same way" in math.

SECTION 8

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

From ISEE probability to future math topics

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.

SECTION 9

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.

Question 1: CONCEPTUAL

Event A has a probability of 0.3 and Event B has a probability of 0.7. Which statement is true?

A. Event A is more likely than Event B.
B. Event A and Event B are equally likely.
C. Event B is more likely than Event A.
D. Both events are impossible.

Show answer (C)

The correct answer is C. Since 0.7 > 0.3, Event B is more likely. Choice A is wrong because 0.3 < 0.7, not greater. Choice B is wrong because the two values are not equal. Choice D is wrong because a probability of 0 means impossible, and neither event has a probability of 0.

Question 2: BASIC CALCULATION

A spinner has 8 equal sections: 3 are red, 2 are blue, and 3 are yellow. Which color are you most likely to spin?

A. Blue
B. Red only
C. Yellow only
D. Red and yellow are equally likely, and both are more likely than blue

Show answer (D)

The correct answer is D. P(red) = 3/8, P(blue) = 2/8, and P(yellow) = 3/8. Since 3/8 > 2/8, both red and yellow are more likely than blue. And since 3/8 = 3/8, red and yellow are equally likely. Choice A is wrong because blue has the lowest probability (2/8). Choices B and C are each wrong because they suggest one color alone is most likely, but red and yellow share the same probability.

Question 3: INTERMEDIATE

A bag contains 4 red, 6 green, and 5 blue marbles. What is the probability of drawing a green marble, and is it greater or less than 1/2?

A. 2/5, which is less than 1/2
B. 2/5, which is greater than 1/2
C. 6/10, which is greater than 1/2
D. 6/15, which is equal to 1/2

Show answer (A)

The correct answer is A. Total marbles = 4 + 6 + 5 = 15. P(green) = 6/15 = 2/5. To compare with 1/2, convert: 2/5 = 4/10 and 1/2 = 5/10. Since 4/10 < 5/10, the probability 2/5 is less than 1/2. Choice B gives the right fraction but the wrong comparison. Choice C incorrectly uses a total of 10 instead of 15. Choice D gives the unsimplified fraction but incorrectly says it equals 1/2.

Question 4: APPLIED

At a school raffle, there are 200 tickets. Mia bought 15 tickets and Jake bought 12 tickets. The probability of drawing a winning ticket that belongs to neither Mia nor Jake is:

A. 12/200
B. 15/200
C. 27/200
D. 173/200

Show answer (D)

The correct answer is D. Mia and Jake together hold 15 + 12 = 27 tickets. The remaining tickets are 200 − 27 = 173. So the probability that the winner is someone else is 173/200. Choice A (12/200) is Jake's probability of winning. Choice B (15/200) is Mia's probability. Choice C (27/200) is the probability that either Mia or Jake wins, which is the opposite of what the question asks.

Question 5: CRITICAL THINKING

A standard deck of 52 cards has 4 aces and 12 face cards (Jacks, Queens, Kings). Which event has a greater probability: drawing an ace or drawing a face card?

A. Drawing an ace, because aces are the highest cards
B. Drawing a face card, because 12/52 > 4/52
C. They are equally likely because both are special cards
D. Drawing an ace, because there are fewer aces so each one is worth more

Show answer (B)

The correct answer is B. P(ace) = 4/52 and P(face card) = 12/52. Since both fractions have the same denominator, just compare numerators: 12 > 4, so drawing a face card is more likely. Choice A is wrong because how "high" a card ranks has nothing to do with probability — only the count matters. Choice C is wrong because the two groups have different numbers of cards. Choice D makes the mistake of confusing rarity with higher probability. Fewer cards means a lower probability, not a higher one.

SUMMARY

Lesson Summary

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).