Probability Rules
Master fundamental probability rules. Learn the addition rule, multiplication rule, complement rule, and complex calculations.
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Fundamental Probability Rules
Before diving into complex probability problems, you need to master three fundamental rules that form the foundation of all probability calculations.
The Complement Rule
The probability that event A does NOT occur equals 1 minus the probability it does occur.
Problem: A die is rolled. What’s the probability of NOT getting a 6?
Solution:
- P(getting 6) = 1/6
- P(NOT getting 6) = 1 - 1/6 = 5/6
Problem: Flip a coin 3 times. What’s the probability of getting at least one head?
Direct approach (complicated):
- P(1 head) + P(2 heads) + P(3 heads)
Complement approach (easier):
- P(at least 1 head) = 1 - P(no heads)
- P(no heads) = P(TTT) = (1/2)³ = 1/8
- P(at least 1 head) = 1 - 1/8 = 7/8
The Addition Rule
The addition rule calculates the probability that at least one of two events occurs (A OR B).
For Mutually Exclusive Events
Events are mutually exclusive if they cannot occur together.
Problem: A card is drawn. What’s P(King or Queen)?
Kings and Queens are mutually exclusive (a card can’t be both).
- P(King) = 4/52
- P(Queen) = 4/52
- P(King or Queen) = 4/52 + 4/52 = 8/52 = 2/13
For Non-Mutually Exclusive Events
Subtract the intersection to avoid double-counting!
Problem: A card is drawn. What’s P(King or Heart)?
These are NOT mutually exclusive—the King of Hearts is both!
- P(King) = 4/52
- P(Heart) = 13/52
- P(King AND Heart) = 1/52 (King of Hearts)
- P(King or Heart) = 4/52 + 13/52 - 1/52 = 16/52 = 4/13
The Multiplication Rule
The multiplication rule calculates the probability that both events occur (A AND B).
For Independent Events
Events are independent if one doesn’t affect the other.
Problem: Flip a coin and roll a die. What’s P(Heads AND 6)?
The coin flip doesn’t affect the die roll.
- P(Heads) = 1/2
- P(6) = 1/6
- P(Heads AND 6) = 1/2 × 1/6 = 1/12
For Dependent Events
Where P(B|A) is the probability of B given that A occurred.
Problem: Draw 2 cards without replacement. What’s P(both Kings)?
After drawing the first King, the deck changes!
- P(1st King) = 4/52
- P(2nd King | 1st was King) = 3/51 (3 Kings left in 51 cards)
- P(both Kings) = 4/52 × 3/51 = 12/2652 = 1/221
Summary Table
| Rule | Formula | When to Use |
|---|---|---|
| Complement | P(Aᶜ) = 1 - P(A) | “NOT” or “at least one” |
| Addition (exclusive) | P(A or B) = P(A) + P(B) | “OR” with no overlap |
| Addition (general) | P(A or B) = P(A) + P(B) - P(A∩B) | “OR” with possible overlap |
| Multiplication (independent) | P(A and B) = P(A) × P(B) | “AND” when independent |
| Multiplication (general) | P(A and B) = P(A) × P(B|A) | “AND” when dependent |
Independence vs Mutually Exclusive
Combining Rules
Complex problems often require combining multiple rules.
Problem: Roll a die twice. What’s P(at least one 6)?
Using complement rule:
- P(at least one 6) = 1 - P(no sixes)
Using multiplication rule (independent rolls):
- P(not 6 on first) = 5/6
- P(not 6 on second) = 5/6
- P(no sixes) = 5/6 × 5/6 = 25/36
Final answer:
- P(at least one 6) = 1 - 25/36 = 11/36
Problem: A bag has 3 red, 2 blue, and 5 green marbles. Draw one. What’s P(red or blue)?
Red and blue are mutually exclusive:
- P(red) = 3/10
- P(blue) = 2/10
- P(red or blue) = 3/10 + 2/10 = 5/10 = 1/2
Summary
In this lesson, you learned:
- Complement Rule: P(not A) = 1 - P(A)
- Addition Rule: P(A or B) = P(A) + P(B) - P(A and B)
- Multiplication Rule: P(A and B) = P(A) × P(B|A)
- For independent events: P(A and B) = P(A) × P(B)
- For mutually exclusive events: P(A or B) = P(A) + P(B)
- Mutually exclusive ≠ independent
- Use complement for “at least one” problems
Practice Problems
1. P(A) = 0.4. What is P(not A)?
2. P(A) = 0.3, P(B) = 0.5, P(A and B) = 0.1. Find P(A or B).
3. A coin is flipped 4 times. What’s P(at least one tail)?
4. A bag has 6 red and 4 blue balls. Two are drawn without replacement. What’s P(both red)?
5. Are “rolling a 2” and “rolling an even number” mutually exclusive?
Click to see answers
1. P(not A) = 1 - 0.4 = 0.6
2. P(A or B) = 0.3 + 0.5 - 0.1 = 0.7
3. Using complement:
- P(no tails) = P(HHHH) = (1/2)⁴ = 1/16
- P(at least one tail) = 1 - 1/16 = 15/16
4. Dependent events (no replacement):
- P(1st red) = 6/10
- P(2nd red | 1st red) = 5/9
- P(both red) = 6/10 × 5/9 = 30/90 = 1/3
5. No, they are NOT mutually exclusive.
Rolling a 2 IS rolling an even number. These events can occur together (and always do when you roll a 2).
Next Steps
Continue building your probability knowledge:
- Conditional Probability - Probability given information
- Bayes’ Theorem - Updating probabilities
- Counting Principles - Permutations and combinations
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