Ever wondered how many ways an event can occur? That’s where counting in probability comes in. It’s a powerful tool that helps you calculate possible outcomes before diving into probability formulas.
In this guide, we’ll break down the essentials, explain the product and sum rules, and walk through easy-to-follow examples.
What is Counting in Probability?
Counting is the foundation of probability. It helps you determine how many outcomes meet a particular condition. Once you can count favorable outcomes, calculating probability becomes easy.
Example 1: Counting Dice Outcomes
Suppose you roll a fair 6-sided die:
- Q: How many possible outcomes are there?
A: 6 outcomes → {1, 2, 3, 4, 5, 6} - Q: How many outcomes are greater than 4?
A: 2 outcomes → {5, 6}
Tip: Define your condition, then count only the outcomes that satisfy it.
The Product Rule of Counting (Step Rule)
Definition:
If an experiment has two independent steps:
- Step 1 has m outcomes
- Step 2 has n outcomes
Total outcomes = m × n
Example 2: Choosing Outfits
You have:
- 3 shirts (Red, Blue, Green)
- 2 pants (Black, White)
Q: How many outfit combinations can you make?
A: 3 × 2 = 6 combinations
Combinations: Red-Black, Red-White, Blue-Black, Blue-White, Green-Black, Green-White
Sum Rule of Counting
Definition:
If an outcome can come from either of two mutually exclusive events:
Total outcomes = m + n
Example 3: Choosing Snacks
A vending machine offers:
- 4 chips
- 3 chocolate bars
Q: How many different snacks can you choose if you pick either one chip or one bar?
A: 4 + 3 = 7 snacks
How Many Valid Bit Strings?
Problem:
A 6-bit string is valid if it:
- Starts with
01OR - Ends with
10
Solution Step-by-Step:
- Total 6-bit strings: 2⁶ = 64
- Start with “01”: Remaining 4 bits → 2⁴ = 16
- End with “10”: First 4 bits → 2⁴ = 16
- Both start with “01” and end with “10”: Middle 2 bits → 2² = 4
Valid strings = 16 + 16 – 4 = 28
Inclusion-Exclusion Rule
Definition:
To avoid double-counting overlapping sets:
|A ∪ B| = |A| + |B| – |A ∩ B|
Example 4: Tea and Coffee Lovers
- 10 people like tea
- 8 like coffee
- 3 like both
Q: How many people like either tea or coffee?
A: 10 + 8 – 3 = 15
Quick Recap: Core Counting Rules
| Rule | Formula | Use Case |
|---|---|---|
| Product Rule | m × n | For sequential, independent steps |
| Sum Rule | m + n | For disjoint (exclusive) choices |
| Inclusion-Exclusion | |A ∪ B| = |A| + |B| – |A ∩ B| | When sets may overlap |
Conclusion
Counting isn’t just math — it’s a practical tool for navigating uncertainty. From games to networks, knowing how to count helps you understand what’s likely — and what’s not.
Up Next: We’ll explore permutations and combinations — the next step in mastering probability.
Permutations and Combinations
Got Questions?
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