Study Tips
How to Understand Probability Properly
By Sana Iqbal · · 4 min read
Quick answer
Probability measures how likely something is, from 0 to 1. Multiply probabilities for 'and' (both events happen), add them for mutually exclusive 'or'. Tree diagrams handle multi-step problems, and the key skill is deciding whether events are independent and whether they can overlap.
The scale from 0 to 1
Every probability sits between 0 (impossible) and 1 (certain). If your answer is negative or above 1, you have made an error — a useful check that catches many mistakes instantly.
Probabilities of all possible outcomes add up to 1, which is why the chance of something not happening is 1 minus the chance it does. That single relationship solves a surprising number of questions.
'And' means multiply
When you want two things to both happen, multiply their probabilities — provided the events are independent. The chance of two coins both landing heads is one half times one half.
The word 'and' in a probability question is usually the signal to multiply. Reading the question for that word is half the battle.
'Or' means add — carefully
For mutually exclusive events — ones that cannot both happen — 'or' means add. The chance of rolling a 1 or a 2 is one sixth plus one sixth.
If the events can overlap, you must subtract the overlap to avoid double-counting. This is the single most common probability error, and it is why 'mutually exclusive' matters so much.
Independent versus dependent
Events are independent when one does not affect the other — two separate coin flips. They are dependent when it does, such as drawing cards without replacing them, where each draw changes what remains.
For dependent events, the second probability must reflect what has already happened. Forgetting to update it is a frequent mistake in 'without replacement' problems.
Tree diagrams organise the thinking
For multi-step problems, a tree diagram lays out every path with its probability. Multiply along each branch, then add the branches that satisfy the question.
Drawing the tree is slower than trying to do it in your head and far more reliable. Under exam pressure, the tree is almost always worth the time.
Trust the rules over intuition
Human intuition about probability is famously poor, which is why the subject feels counter-intuitive. The rules exist precisely because guessing does not work.
When your gut disagrees with a careful calculation, the calculation is almost always right. Learning to trust the method over the hunch is much of what the topic teaches.
Frequently asked questions
When do I add and when do I multiply probabilities?+
Multiply for 'and' — when you want both events to happen. Add for 'or' with mutually exclusive events. If events can overlap, subtract the overlap to avoid double-counting.
What does independent mean?+
Two events are independent if one happening does not change the probability of the other, like separate coin flips. Dependent events, such as drawing cards without replacing them, affect each other.
Why are tree diagrams useful?+
They lay out every possible path in a multi-step problem with its probability, so you multiply along branches and add the relevant ones. They make organised what is easy to get wrong in your head.
How do I find the probability of something not happening?+
Subtract the probability that it does happen from 1, because all outcomes together add up to 1. This is often much faster than adding up every other possibility.
Why does probability feel so counter-intuitive?+
Human intuition about chance is genuinely unreliable, which is why the subject uses strict rules. When your instinct disagrees with a careful calculation, trust the calculation.
Try your first session free
Meet your tutor, set your goals, and see the difference one-to-one attention makes. No card required, no commitment.