Probability Formula- Basic Rules and Examples
What Is Probability?
Probability measures how likely something is to happen. It's expressed as a number between 0 and 1. Zero means impossible. One means certain. Everything in between covers the gray area of real life.
You use probability every day without realizing it. Will it rain tomorrow? Should you take an umbrella? That's probability working in your head.
The Basic Probability Formula
Here's the core formula you need to know:
P(A) = Number of favorable outcomes / Total number of possible outcomes
That's it. Divide what you want by what's possible.
Example: What's the probability of rolling a 4 on a fair six-sided die?
- Favorable outcomes: 1 (only one face shows 4)
- Total outcomes: 6 (six faces total)
- P(4) = 1/6 ≈ 0.167 or 16.7%
Key Probability Rules You Must Know
The Addition Rule
Use this when you want the probability of either event A or event B happening.
P(A or B) = P(A) + P(B) - P(A and B)
The subtraction handles overlap so you don't count the same outcome twice.
Example: Drawing a King or a Heart from a standard deck.
- P(King) = 4/52
- P(Heart) = 13/52
- P(King of Hearts) = 1/52
- P(King or Heart) = 4/52 + 13/52 - 1/52 = 16/52 ≈ 30.8%
The Multiplication Rule
Use this when you want the probability of both events happening.
For independent events (one doesn't affect the other):
P(A and B) = P(A) × P(B)
Example: Flipping a heads and rolling a 6. These are independent.
- P(Heads) = 1/2
- P(6) = 1/6
- P(Heads AND 6) = 1/2 × 1/6 = 1/12 ≈ 8.3%
For dependent events, you need conditional probability. The second event's probability changes based on what happened first.
P(A and B) = P(A) × P(B|A)
The Complement Rule
The complement is "not A" — everything except what you're measuring.
P(not A) = 1 - P(A)
Example: What's the probability of NOT rolling a 3?
- P(3) = 1/6
- P(not 3) = 1 - 1/6 = 5/6 ≈ 83.3%
This rule is useful when counting "at least one" scenarios.
Types of Probability
| Type | Description | Example |
|---|---|---|
| Theoretical | Based on logic and structure | Fair coin = 50% chance heads |
| Experimental | Based on actual trials | Flip coin 100 times, get 48 heads |
| Axiomatic | Mathematical foundation (Kolmogorov) | Three axioms govern all probability |
Common Probability Mistakes to Avoid
- Ignoring order when it matters: "AB" and "BA" are different if order counts
- Forgetting to adjust for dependent events: The second draw changes probabilities in a deck
- Confusing "or" with "and": Or means either. And means both.
- Assuming equally likely outcomes: Real-world situations aren't always fair
How to Calculate Probability: Step-by-Step
Step 1: Define your event clearly.
Step 2: Count total possible outcomes.
Step 3: Count favorable outcomes (what you want).
Step 4: Apply the formula: favorable ÷ total.
Step 5: Simplify your fraction if needed.
Step 6: Convert to percentage or decimal based on context.
Quick Reference Table
| Scenario | Formula | When to Use |
|---|---|---|
| Single event | P(A) = f/n | Basic probability questions |
| Either event | P(A∪B) = P(A) + P(B) - P(A∩B) | At least one condition met |
| Both events | P(A∩B) = P(A) × P(B) | Independent events both occur |
| Not this event | P(A') = 1 - P(A) | Finding the opposite outcome |
Real Example: Birthday Problem
What's the probability that two people in a room share a birthday? 🎂
It's easier to find the probability they don't share one, then subtract from 1.
- Person 1: 365/365 (any birthday)
- Person 2: 364/365 (not Person 1's birthday)
- Person 3: 363/365
- Continue for n people
P(no match) = (365 × 364 × 363 × ... × (365-n+1)) / 365ⁿ
With just 23 people, P(match) exceeds 50%. Counterintuitive but true.
When Probability Gets Complicated
For complex problems, break them into smaller pieces. Ask:
- Are the events independent?
- Does order matter?
- What am I actually counting?
Draw a tree diagram if you need to. Visualize the outcomes. Most probability errors come from trying to solve in your head instead of writing it out.