If you toss a coin and roll a die at the same time, what’s the chance of getting a head and a 6?
Welcome to the world of Probability Laws where we don't just guess outcomes, we calculate them with logic.
This article will help you understand the three key rules that govern probability: the Addition Law, the Multiplication Law, and the Complementary Rule. These are essential tools for solving probability problems accurately in exams and real-world scenarios.
Addition Law of Probability
When you want to know the probability of either one event or another happening, you apply the addition law. But first, ask: Are the events mutually exclusive?
(a) Mutually Exclusive Events
Events that cannot happen at the same time such as getting a 2 or a 5 on a single die roll.
Formula:
P(A or B) = P(A) + P(B) (only if A and B are mutually exclusive)
Example: Probability of drawing a King or a Queen from a deck: P(K) = 4/52, P(Q) = 4/52 → P(K or Q) = 4/52 + 4/52 = 8/52
(b) Non-Mutually Exclusive Events
Events that can occur simultaneously like Drawing a red card or a Queen.
Formula:
P(A or B) = P(A) + P(B) – P(A and B)
Example: P(Red) = 26/52, P(Queen) = 4/52, P(Red Queen) = 2/52 → Total = 26/52 + 4/52 – 2/52 = 28/52
Multiplication Law of Probability
This rule answers the question: What’s the chance that both events A and B happen?
(a) Independent Events
Two events are independent if the occurrence of one doesn’t affect the other. Example: Tossing a coin and rolling a die.
Formula:
P(A and B) = P(A) × P(B)
Example: P(Head) = 1/2, P(6) = 1/6 → P(Head and 6) = 1/2 × 1/6 = 1/12
(b) Dependent Events
Events where one affects the probability of the other such as Drawing two cards without replacement.
Formula:
P(A and B) = P(A) × P(B|A) where P(B|A) is the probability of B given A has occurred.
Example:
First card is an Ace: 4/52
Second card is Ace (after removing 1): 3/51 → P(Ace and Ace) = 4/52 × 3/51 = 12/2652
Complementary Rule
Sometimes it's easier to calculate the probability that something won’t happen and then subtract it from 1.
Formula:
P(A') = 1 – P(A)
Example: The probability of not getting a 6 on a die roll = 1 – 1/6 = 5/6
This rule is especially useful when the "not" event is simpler to evaluate than the event itself.
Examples for Practice
Example 1: Addition Rule (Mutually Exclusive)
What is the probability of drawing either a 5 or a 6 from a fair die?
P(5) = 1/6, P(6) = 1/6 → P(5 or 6) = 1/6 + 1/6 = 2/6 = 1/3
Example 2: Multiplication Rule (Independent)
Probability of getting a head and an even number (on die)?
P(H) = 1/2, P(Even) = 3/6 = 1/2 → P(H and Even) = 1/2 × 1/2 = 1/4
Example 3: Complementary Rule
Probability of not getting a King from a deck of cards?
P(King) = 4/52 → P(Not King) = 1 – 4/52 = 48/52
Quick Recap
| Rule | When to Use | Formula | Example Outcome |
|---|---|---|---|
| Addition (Mutually Exclusive) | One event OR another (no overlap) | P(A or B) = P(A) + P(B) | Getting red or black card → 0.5 + 0.5 = 1 |
| Addition (Not Mutually Exclusive) | One event OR another (with overlap) | P(A or B) = P(A) + P(B) – P(A and B) | Red or Queen → 26/52 + 4/52 – 2/52 = 28/52 |
| Multiplication (Independent) | Both events happen independently | P(A and B) = P(A) × P(B) | Head and 6 → 1/2 × 1/6 = 1/12 |
| Multiplication (Dependent) | Second depends on first | P(A and B) = P(A) × P(B|A) | Two aces drawn → 4/52 × 3/51 |
| Complement Rule | Something does not happen | P(A') = 1 – P(A) | Not drawing a King → 1 – 4/52 = 48/52 |
Why Do These Laws Matter?
Without them, you'd be guessing. With them, you're thinking like a statistician.
Probability laws aren’t just for dice and cards. They're essential for forecasting sales, planning marketing strategies, managing inventory, calculating risk, and even diagnosing diseases.
Do you rely on luck, or logic? These rules are the bridge between randomness and reasoning.
Start seeing events around you missed buses, rainy days, online orders through the lens of probability. It’s not magic. It’s math with purpose.