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Basic
Probability

Learn to calculate how likely an event is to happen using the probability formula and apply it to real-world scenarios.

ISEE Middle Level

What Is Probability?

Probability is a number that tells us how likely something is to happen. It answers the question: "What are the chances?"

Certain
The sun will rise tomorrow: probability = \(1\)
Possible
Rolling a \(3\) on a die: probability between \(0\) and \(1\)
Impossible
Rolling a \(7\) on a standard die: probability = \(0\)

Key Vocabulary

1
Experiment: An action that produces outcomes (e.g., flipping a coin).
2
Outcome: A single possible result (e.g., getting Heads).
3
Sample Space: The set of all possible outcomes.
4
Event: A specific outcome or group of outcomes you care about.
5
Favorable Outcomes: Outcomes that match the event you want.

The Probability Scale

0 Impossible 0.25 Unlikely 0.5 Even Chance 0.75 Likely 1 Certain

Probability can be written as a fraction, decimal, or percent. For example, \(\frac{1}{2} = 0.5 = 50\%\).

The Probability Formula

$$P(\text{Event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}}$$

Formula In Action

Bag of Marbles R R R B B G Total: 6 marbles 3 Red 2 Blue 1 Green

Theoretical vs. Experimental

Theoretical
What should happen based on math. Uses the formula with equally likely outcomes.
Experimental
What actually happens when you run an experiment. Based on observed data.

Complementary Events

The complement of an event is everything that is not that event. Together, an event and its complement always add up to \(1\).

Event A P(A) Not A P(not A) P(A) + P(not A) = 1

How to Solve Any Problem

1
Identify the experiment. What action is being performed? (rolling a die, picking a card, etc.)
2
List the sample space. Count all possible outcomes (the denominator).
3
Count favorable outcomes. How many outcomes match the event? (the numerator)
4
Write the fraction. \(P = \frac{\text{favorable}}{\text{total}}\) and simplify if possible.
5
Check your answer. Is it between \(0\) and \(1\)? Does it make sense?

Example 1: Coin Flip

Problem: You flip a fair coin. What is the probability of getting Heads?

H Heads T Tails Sample Space: {H, T}

Example 2: Rolling a Die

Problem: You roll a standard six-sided die. What is the probability of rolling an even number?

1 2 3 4 5 6 Green = Even (favorable) Grey = Odd (not favorable)

Example 3: Spinner Problem

Problem: A spinner has 4 red, 3 blue, and 1 yellow equal sections. What is the probability of landing on blue?

R R R R B B B Y

Example 4: Using the Complement

Problem: A bag contains 10 candies: 2 are green. What is the probability of picking a candy that is not green?

1
Find \(P(\text{Green})\): \(\frac{2}{10} = \frac{1}{5}\)
2
Use the complement rule: \(P(\text{not Green}) = 1 - P(\text{Green})\)
3
Calculate: \(P(\text{not Green}) = 1 - \frac{1}{5} = \frac{4}{5}\)

Common Mistakes to Avoid

✕ Forgetting to Simplify
Writing \(\frac{3}{6}\) instead of \(\frac{1}{2}\). Always reduce your fraction to lowest terms.
✕ Probability Over 1
Getting a result like \(\frac{5}{3}\). This is impossible: probability can never exceed \(1\). Recheck your counts.
✕ Mixing Up Numerator
Putting the total on top and favorable on bottom. Remember: favorable goes on top.

Probability Cheat Sheet

Concept Formula Example
Basic Probability \(\frac{\text{favorable}}{\text{total}}\) \(\frac{3}{6} = \frac{1}{2}\)
Complement \(1 - P(A)\) \(1 - \frac{1}{4} = \frac{3}{4}\)
Impossible Event \(P = 0\) Rolling 7 on a die
Certain Event \(P = 1\) Getting H or T on a flip
As Percent \(P \times 100\%\) \(\frac{1}{4} = 25\%\)

Question 1 of 5

A jar contains 5 red, 3 blue, and 2 yellow marbles. If you pick one marble at random, what is the probability of picking a blue marble?

Question 2 of 5

The probability of rain tomorrow is \(\frac{2}{5}\). What is the probability that it will not rain?

Question 3 of 5

A standard deck has 52 cards. There are 4 aces. What is the probability of drawing an ace?

Question 4 of 5

A spinner has 8 equal sections numbered \(1\) through \(8\). What is the probability of landing on a number greater than 5?

Question 5 of 5

You roll a standard die. What is the probability of rolling a number that is both even and greater than 3?