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Calculating
Probability

Learn how to find the chance of an event using counts and ratios with the formula \( P = \dfrac{\text{favorable}}{\text{total}} \).

ISEE Quantitative Reasoning

What Is Probability?

Probability tells us how likely something is to happen. It is a number that measures the chance of an event.

The Probability Scale

0
\(\frac{1}{2}\)
1 Impossible Equal chance Certain More likely →

Key Words to Know

Outcome
A single possible result, like rolling a 4 on a die.
Event
What we want to happen, like "rolling an even number."
Favorable
The outcomes that count as success for our event.

The Probability Formula

To find probability, count the outcomes you want (favorable), then count all the possible outcomes (total). Put the first number over the second.

Counting With a Bag of Marbles

\(3\) green + \(2\) yellow = \(5\) marbles

Favorable vs. Total

It is easy to mix up the two numbers in the fraction. Remember which goes on top and which goes on the bottom.

Top (Numerator)
Only the outcomes you want. The favorable count.
Bottom (Denominator)
All the outcomes that could happen. The total count.

How to Solve a Probability Problem

1
Find the total. Count how many outcomes are possible in all.
2
Find the favorable. Count how many outcomes match the event.
3
Write the fraction. Put favorable over total: \( \dfrac{\text{favorable}}{\text{total}} \).
4
Simplify the fraction if you can.

Example 1: Rolling a Die

1 2 3 4 5 6 Even numbers shown in green

Example 2: Picking a Card

A box has 4 red cards, 3 blue cards, and 1 green card. What is the chance of picking a blue card?

1
Total cards: \( 4 + 3 + 1 = 8 \) cards.
2
Favorable: there are \( 3 \) blue cards.
3
Probability: \( P(\text{blue}) = \dfrac{3}{8} \).

Example 3: Simplifying the Answer

A spinner has 10 equal sections. 4 of them are orange. What is the probability of landing on orange?

Always simplify the fraction at the end. \( \dfrac{2}{5} \) is the cleaner, final answer.

Common Mistakes

✕ Flipping the fraction
Don't put total on top. Favorable goes on top, total on bottom.
✕ Forgetting to add
The total must include every outcome, not just the ones you want.
✕ Not simplifying
Answers like \( \frac{4}{10} \) should be reduced to \( \frac{2}{5} \).

Quick Summary

Favorable Total
Where in fractionTop (numerator)Bottom (denominator)
What it countsOutcomes you wantAll outcomes
Roll even on die36
Result\( \frac{3}{6} = \frac{1}{2} \)

You're Ready!

Count what you want, count everything, write the fraction, and simplify. Now let's practice with some questions!

Question 1 of 5

A bag has 2 red and 3 blue marbles. What is the probability of picking a red marble?

Question 2 of 5

A fair die is rolled. What is the probability of rolling a number greater than 4?

Question 3 of 5

A spinner has 8 equal sections. 2 are purple. What is the probability of landing on purple, in simplest form?

Question 4 of 5

A jar has 5 green, 4 red, and 3 yellow candies. What is the probability of picking a yellow candy?

Question 5 of 5

There are 20 students in a class. 12 of them have a pet. What is the probability that a randomly chosen student has a pet?

Great Work!

You now know how to calculate probability from counts and ratios. Remember: favorable over total, then simplify. Keep practicing!