Comparing Probabilities — Compare probabilities of two...

SECTION 1

Historical Context & Motivation

People have been comparing chances for thousands of years. Ancient civilizations rolled dice, flipped coins, and played card games. They noticed that some outcomes happened more often than others. Over time, mathematicians turned these observations into a real branch of math called probability (the study of how likely events are to happen).

~3000 BCE

Ancient Dice Games

People in Mesopotamia used bone dice for games. They noticed some rolls came up more often than others.

1654

Pascal & Fermat Lay the Foundation

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

1713

Bernoulli's Ars Conjectandi

Jakob Bernoulli published a book that gave us the rules for calculating and comparing probabilities. This made probability a formal part of mathematics.

Today

Probability on the SHSAT

Comparing probabilities is a key skill tested on the SHSAT. You need to figure out which event is more, less, or equally likely.

On the SHSAT, you will often see two events described in a problem. Your job is to find each probability and then decide which one is greater. This lesson will teach you exactly how to do that.

SECTION 2

Core Principles & Definitions

Before you can compare probabilities, you need to understand a few key ideas. Let's break them down one at a time.

Probability of an Event

Probability tells you how likely something is to happen. It is always a number from 0 (impossible) to 1 (certain). You can also write it as a fraction, decimal, or percent.

Favorable Outcomes

Favorable outcomes are the specific results you are looking for. For example, if you want to roll a 3 on a die, there is 1 favorable outcome.

Total Outcomes

Total outcomes means every possible result. A standard die has 6 total outcomes: 1, 2, 3, 4, 5, and 6.

Common Denominator

To compare two fractions, you often need a common denominator (the same bottom number). This makes it easy to see which fraction is bigger.

Equally Likely Outcomes

When every outcome has the same chance of happening, we say the outcomes are equally likely. A fair coin and a fair die both have equally likely outcomes.

✦ KEY TAKEAWAY

Think of probability like a volume knob on a speaker. At 0, there is no sound at all (impossible). At 1, the sound is at maximum (certain). Comparing probabilities is like checking which of two speakers is turned up louder.

SECTION 3

Visual Explanation — Probability Number Line

A great way to compare probabilities is to place them on a number line from 0 to 1. The event that is farther to the right is more likely. Let's look at an example.

The number line shows probabilities from 0 (impossible) to 1 (certain). Event A sits at 1/4, while Event B sits at 7/12. Since Event B is farther to the right, it is more likely.

Whenever you place two probabilities on a number line, the one that lands farther to the right has a higher chance of happening. This is one of the quickest ways to compare events at a glance.

SECTION 4

Mathematical Framework

To compare two probabilities on the SHSAT, you first need to calculate each one. Here is the basic formula.

PROBABILITY FORMULA

P(event) = favorable outcomes ÷ total outcomes

P(event) = the probability of the event happening. Favorable outcomes = how many results you want. Total outcomes = every possible result.

Once you have both probabilities as fractions, you can compare them. There are three handy methods.

METHOD 1 — COMMON DENOMINATORS

Rewrite both fractions with the same denominator, then compare numerators.

Example: Compare 2/5 and 3/8. The least common denominator is 40, so 2/5 = 16/40 and 3/8 = 15/40. Since 16 > 15, we know 2/5 > 3/8.

METHOD 2 — CONVERT TO DECIMALS

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

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

METHOD 3 — CROSS MULTIPLICATION

For a/b vs. c/d, compare a × d with c × b.

Example: Compare 2/5 and 3/8. Compute 2 × 8 = 16 and 3 × 5 = 15. Since 16 > 15, we know 2/5 > 3/8. The side with the larger product wins.

💡 SHSAT Tip

Cross multiplication is usually the fastest method on a timed test. You skip finding a common denominator and go straight to comparing two products.

SECTION 5

Detailed Breakdown — Three Comparison Methods

Let's see all three methods side by side using a visual chart. Suppose a bag has 3 red marbles and 5 blue marbles (8 total). You want to compare the probability of picking red versus the probability of picking blue.

This chart shows all three comparison methods applied to the same problem. A bag contains 3 red and 5 blue marbles. All three methods confirm that P(Blue) > P(Red).

When the fractions already share the same denominator (like 3/8 and 5/8), you can skip straight to comparing the numerators. When the denominators are different, pick whichever method feels fastest to you. On the SHSAT, cross multiplication is usually the quickest because you only multiply two pairs of numbers.

SECTION 6

Worked Example

Let's walk through a full SHSAT-style problem step by step.

📝 Problem

A spinner is divided into 10 equal sections numbered 1 through 10. Event A is landing on an even number. Event B is landing on a number less than 4. Which event has a greater probability?

Step-by-Step Solution

Step 1 — List the total outcomesThe spinner has 10 equal sections, so there are 10 total outcomes: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

Total outcomes = 10

Step 2 — Find favorable outcomes for Event AEvent A is landing on an even number. The even numbers from 1 to 10 are: {2, 4, 6, 8, 10}. That gives us 5 favorable outcomes.

P(A) = 5/10 = 1/2

Step 3 — Find favorable outcomes for Event BEvent B is landing on a number less than 4. The numbers less than 4 are: {1, 2, 3}. That gives us 3 favorable outcomes.

P(B) = 3/10

Step 4 — Compare the two probabilitiesBoth fractions already have a denominator of 10. Compare the numerators: 5 > 3. So P(A) > P(B).

Event A (even number) is more likely than Event B (number less than 4).

Notice how we used the same denominator (10) because both events use the same spinner. When the total outcomes are the same, you only need to compare the number of favorable outcomes. More favorable outcomes means a higher probability!

SECTION 7

Strengths, Pitfalls, and Comparison Tips

Each comparison method has its strengths and weaknesses. Here is a quick guide to help you pick the right tool on test day.

Comparison method cheat sheet

Method	Best When…	Watch Out For…
Common Denominator	Denominators are small or already the same. Easy to find the LCD quickly.	Large denominators can make this slow. Don't forget to adjust both numerators.
Convert to Decimals	Denominators like 4, 5, 8, 10, or 20 that divide neatly.	Repeating decimals (like 1/3 = 0.333…) can be tricky without a calculator.
Cross Multiplication	You want speed. Works with any two fractions, no need to find an LCD.	Keep track of which product goes with which fraction. The left product matches the left fraction.

✦ KEY TAKEAWAY

Think of comparing probabilities like comparing prices at two stores. You can convert both prices to the same currency (common denominator), look at the dollar amounts (decimals), or use a quick shortcut (cross multiply). The answer is the same no matter which way you check.

⚠️ Common SHSAT Trap

Some problems give probabilities in mixed forms — one as a fraction and one as a percent. Always convert them to the same form before comparing. For example, 3/5 = 60%, so you can compare it directly to 55%.

SECTION 8

Connection to Advanced Probability

Comparing simple probabilities is a stepping stone to more advanced topics you will see in high school and beyond. Here is how today's skill connects to what comes next.

From basic to advanced probability

What You Learn Now	What Comes Next
Compare P(A) and P(B) for simple events (one die, one spinner, one bag of marbles).	Compare probabilities of compound events (rolling two dice, picking two cards).
Use fractions, decimals, and percents to express probability.	Use probability notation like P(A ∩ B) for the probability of A and B both happening.
Decide which event is more likely.	Calculate expected value — how much you expect to win or lose over many tries.
Count favorable and total outcomes by listing them.	Use combinations and permutations to count outcomes in complex situations.

The good news is that the core idea never changes. No matter how complicated the problem gets, you always find the probability of each event and then compare. Master this skill now, and the advanced topics will feel much easier later.

SECTION 9

Practice Problems

Try these five problems on your own. They get harder as you go. Check your answers after each one!

PROBLEM 1 — CONCEPTUAL

Event X has a probability of 0.6, and Event Y has a probability of 3/4. Without calculating, explain how you could decide which event is more likely.

PROBLEM 2 — BASIC CALCULATION

A bag contains 4 green marbles, 6 yellow marbles, and 2 red marbles. Compare the probability of picking a green marble to the probability of picking a red marble.

PROBLEM 3 — INTERMEDIATE

A standard deck of 52 cards is shuffled. Event A is drawing a heart. Event B is drawing a face card (Jack, Queen, or King of any suit). Which event is more likely?

PROBLEM 4 — APPLIED

A game show has two wheels. Wheel 1 has 8 equal sections, and 3 of them say "WIN." Wheel 2 has 12 equal sections, and 5 of them say "WIN." You get to spin only one wheel. Which wheel gives you a better chance of winning?

PROBLEM 5 — CRITICAL THINKING

Ava says that since a coin has 2 outcomes and a die has 6 outcomes, flipping heads on a coin (P = 1/2) is "three times as likely" as rolling a 6 on a die (P = 1/6). Is she correct? Explain your reasoning, and find the actual ratio of the two probabilities.

SUMMARY

Lesson Summary

To compare the probabilities of two events, start by calculating each probability using the formula P(event) = favorable outcomes ÷ total outcomes. Then express both probabilities in the same form — as fractions with a common denominator, as decimals, or by using cross multiplication. The event with the larger value is more likely to happen.

Remember that all probabilities fall between 0 (impossible) and 1 (certain). On the SHSAT, watch out for problems that mix fractions, decimals, and percents — always convert to the same form first. When the total outcomes are the same for both events, you only need to compare the number of favorable outcomes. Practice these skills until comparing probabilities feels automatic!