Permutations and Combinations in Probability (With Examples)

Statistics & Probability Made Simple (Step-by-Step)

📚 Table of Contents

• How to Think About Probability
• Marbles Example
• Combination Calculator
• Probability Tree
• Dice
• Coin
• Key Takeaways

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🧠 How to Think About Probability

Before formulas, understand this:

💡 Probability = "How likely something is to happen"

Simple rule:

Probability = Favorable outcomes / Total outcomes

Example: If 1 red ball exists out of 5 → chance = 1/5

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🎱 Marbles (Step-by-Step Thinking)

👉 Step 1: Ask This Question

Does order matter?

• Yes → Permutation
• No → Combination

🔢 Permutation (Order Matters)

Example: Picking winners (1st, 2nd place)

P(5,2) = 20

📊 Combination (Order Doesn't Matter)

Example: Selecting a team

C(5,2) = 10

🎯 Probability (Step-by-Step)

We want: Red → then Blue

💡 Multiply probabilities when events happen in sequence

(1/5) × (1/4) = 1/20

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🧮 Interactive Combination Calculator

Enter n: Enter r:

Result:

💡 Use this when order does NOT matter

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🌳 Probability Tree (Visual Thinking)

Explanation:

• Each branch = probability
• Multiply along a path
• Add different paths if needed

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🎲 Dice

💡 Expected Value (Simple Idea)

This means "average result if you play many times"

(1+2+3+4+5+6)/6 = 3.5

📊 Probability Tip

💡 Always count total possible outcomes first

P(sum=7) = 6/36 = 1/6

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🪙 Coin

📊 Binomial (Simple Thinking)

Used when:

• Fixed number of trials
• Only 2 outcomes

P(3 heads) = 0.3125

📈 Big Idea

💡 More trials → results become stable (50-50)

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🎯 Key Takeaways

✔ Ask: does order matter? ✔ Multiply probabilities for sequences ✔ Count total outcomes carefully ✔ Use combinations for selection ✔ Use permutations for ranking

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🚀 Final Thought

Don't memorize formulas first. Understand the situation → then apply the formula.

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📚 Related Article

• Additive Probability Explained