Three Rules That Unlock Probability
In the previous lesson, we learned that probability is a number between 0 and 1 that measures how likely something is to happen. Now we need some rules for combining probabilities - because real life rarely involves just one event at a time.
The good news: there are only three basic rules you need, and they all follow common sense once you see them in action.
Rule 1: The Complement Rule - "What Are the Chances It Does NOT Happen?"
Sometimes the easiest way to find a probability is to think about the opposite. The complement of an event is everything that is not that event.
The rule is beautifully simple:
P(event does NOT happen) = 1 − P(event happens)
Since all possibilities must add up to 1 (something has to happen), the chance of "not A" is just 1 minus the chance of "A."
The weather forecast says there is a 30% chance of rain today. What is the chance it does NOT rain?
P(no rain) = 1 − 0.30 = 0.70, or 70%.
That is it. If there is a 30% chance of rain, there is a 70% chance of dry weather.
You are playing a board game and need to roll anything except a 1 on a six-sided die to stay in the game. What is the probability you survive?
P(rolling a 1) = 1/6 ≈ 0.167. So P(NOT rolling a 1) = 1 − 1/6 = 5/6 ≈ 0.833, or about 83.3%. The odds are in your favor.
The complement rule is especially handy when there are many ways something can happen and only a few ways it cannot. Instead of adding up all the ways it can happen, just calculate the few ways it cannot - and subtract from 1.
Rule 2: The Addition Rule - "What Are the Chances of This OR That?"
When you want to know the probability of one event or another happening, you use the addition rule. But there is an important detail: can both events happen at the same time?
When Events Cannot Overlap (Mutually Exclusive)
Two events are mutually exclusive if they cannot happen at the same time. When you roll a die, you cannot get both a 3 and a 5 on the same roll. When events are mutually exclusive, you simply add their probabilities:
P(A or B) = P(A) + P(B)
You roll a die. What is the probability of getting a 2 or a 5?
These are mutually exclusive - you cannot roll both at once. P(2) = 1/6 and P(5) = 1/6.
P(2 or 5) = 1/6 + 1/6 = 2/6 = 1/3 ≈ 0.333, or about 33.3%.
When Events CAN Overlap
If two events can happen at the same time, adding their probabilities would count the overlap twice. So you need to subtract it out:
P(A or B) = P(A) + P(B) − P(A and B)
In a standard deck of 52 playing cards, what is the probability of drawing a heart or a queen?
There are 13 hearts and 4 queens. But one card - the queen of hearts - is both. If we just add 13 + 4 = 17, we count that card twice.
P(heart) = 13/52. P(queen) = 4/52. P(queen of hearts) = 1/52.
P(heart or queen) = 13/52 + 4/52 − 1/52 = 16/52 = 4/13 ≈ 0.308, or about 30.8%.
A good way to remember this: if you are combining events with "or," think about whether there is any overlap. If there is, subtract it once so you do not double-count.
Rule 3: The Multiplication Rule - "What Are the Chances of This AND That?"
When you want to know the probability of two events both happening, you multiply. But again, there is a detail that matters: does the first event affect the second?
Independent Events
Two events are independent if one happening does not change the probability of the other. Flipping a coin and rolling a die are independent - the coin does not care what the die does.
P(A and B) = P(A) × P(B)
You flip a coin and roll a die. What is the probability of getting heads AND a 6?
P(heads) = 1/2. P(rolling a 6) = 1/6. These events are independent.
P(heads and 6) = 1/2 × 1/6 = 1/12 ≈ 0.083, or about 8.3%.
Dependent Events
When one event affects the other, they are dependent. In that case, the probability of the second event changes based on what happened first. We will explore this more deeply in the next lesson on conditional probability, but here is a preview:
You have a bag with 5 red and 3 blue marbles (8 total). You draw one marble, do NOT put it back, and draw a second. What is the probability of drawing two red marbles in a row?
First draw: P(red) = 5/8.
Second draw (given the first was red): there are now 4 red and 3 blue left (7 total). P(red) = 4/7.
P(both red) = 5/8 × 4/7 = 20/56 = 5/14 ≈ 0.357, or about 35.7%.
Notice the second probability changed because we removed a marble. That is what makes these events dependent.
Putting the Rules Together in Daily Life
These rules are not just for cards and dice. You use this logic all the time without realizing it.
Shopping Decisions
A store has a 40% chance of having your size in stock, and independently, a 25% chance of having a sale. What is the probability that you find your size AND it is on sale?
P(your size and sale) = 0.40 × 0.25 = 0.10, or 10%. Not great odds - maybe call ahead.
Health Screening
Suppose 2% of people in a certain age group have a condition. The complement rule tells you that 98% do NOT have it. Simple, but powerful - it frames how common or rare something really is.
Travel Planning
Your first flight has a 90% on-time record. Your connection also has a 90% on-time record. If these are independent, the probability that BOTH are on time is 0.90 × 0.90 = 0.81, or 81%. Suddenly a 90% rate does not feel so comfortable when you need two things to go right.
You need three independent things to all go right for a project to succeed. Each has a 95% chance of working. What is the overall probability of success?
P(all three succeed) = 0.95 × 0.95 × 0.95 = 0.857, or about 85.7%.
Even though each step is very likely, stringing them together reduces the overall odds. This is why backup plans matter.
Common Mistakes to Avoid
- Forgetting the overlap in "or" problems. If events can happen at the same time, remember to subtract the overlap.
- Assuming independence. Drawing cards without replacement, for example, creates dependent events. Always ask: "Does the first event change the situation for the second?"
- Adding when you should multiply. "Or" means add (with the overlap adjustment). "And" means multiply. Mixing these up is one of the most common errors.
A Quick Summary of the Three Rules
- Complement: P(not A) = 1 − P(A)
- Addition (or): P(A or B) = P(A) + P(B) − P(A and B)
- Multiplication (and): P(A and B) = P(A) × P(B) if independent
These three rules are the building blocks for everything else in probability. Master them, and you will have a solid foundation for the more advanced topics that follow.
The three basic probability rules cover the most common questions: the complement rule tells you the chance something does NOT happen (subtract from 1), the addition rule handles "or" situations (add probabilities, subtract any overlap), and the multiplication rule handles "and" situations (multiply probabilities, adjusting if events are dependent). Together, these rules let you break complex situations into simple calculations.