Probability Rules: When to Add and When to Multiply

Quick answer: In probability, "and" means multiply and "or" means add. Multiply along the branches of a tree when two things both happen, add separate outcomes that cannot happen together, subtract the overlap when they can, and for any "at least one" question work out 1 minus the probability of none.

I lost four marks on one probability question in my IGCSE mock in January, and it was not because I did not know the rules. It was because I read "two red counters" and wrote 7/10 + 6/9 instead of multiplying. I knew the multiplication rule. I just never asked myself the AND-or-OR question before picking up the pen. Since then I write two letters at the top of every probability question, A or O, and I have not made that mistake again.

Step one: decide whether the question is AND or OR

Before any arithmetic, translate the question into plain English and find the joining word. That single decision controls everything else.

• "Both", "and then", "followed by", "all three" means AND, so multiply.
• "Either", "or", "one of these two named outcomes" means OR, so add.
• "At least one" across several tries means use the complement.
• "Given that", "if we know that" means conditional probability, so the sample space has shrunk.

Most questions mix the two. "Both counters the same colour" is AND inside each branch and OR between them, which is why the method is multiply along, then add down.

The addition rule: OR means add, but subtract the overlap

The general addition rule is P(A or B) = P(A) + P(B) - P(A and B). The subtraction is there because anything sitting in both events gets counted twice if you just add.

Real numbers. In my history class of 30, 18 take history, 14 take geography, and 8 take both. Picking one student at random, the probability they take at least one of the two is 18/30 + 14/30 - 8/30 = 24/30 = 4/5.

Skip the minus 8/30 and you get 32/30, which is bigger than certainty and therefore impossible. Any answer above 1 or below 0 means a structural mistake, not a small slip.

When two events cannot happen at the same time (rolling a 3 and rolling a 5 on one die), the overlap is zero and the rule collapses to P(A) + P(B). These are mutually exclusive events, and Cambridge and Edexcel papers like asking you to say so before you calculate.

The multiplication rule: AND means multiply along the branches

For two events happening together, P(A and B) = P(A) × P(B given A). If the first event does not change the second, the events are independent and it simplifies to P(A) × P(B).

The phrase that decides it is "with replacement" or "without replacement". With replacement, the counter goes back in and every branch of your tree carries the same probabilities. Without replacement, the fraction changes on the second pick: the top changes only if you took one of that colour, but the bottom always drops by one, because there is one fewer object in the bag either way.

Worked example: two counters without replacement

A bag has 7 red counters and 3 blue counters. Two counters are taken out without replacement. Find the probability that they are the same colour.

1. Draw the tree. First pick: red 7/10, blue 3/10.
2. Second branch after red: 6 red left out of 9 total, so red 6/9, blue 3/9.
3. Second branch after blue: 7 red left out of 9 total, so red 7/9, blue 2/9.
4. Multiply along the red-red path: 7/10 × 6/9 = 42/90.
5. Multiply along the blue-blue path: 3/10 × 2/9 = 6/90.
6. Those two paths are the OR part, so add: 42/90 + 6/90 = 48/90 = 8/15.

Now check it a second way, which I do on any question worth 4 or more marks. P(different colours) should be 1 - 8/15 = 7/15. Directly: 7/10 × 3/9 + 3/10 × 7/9 = 21/90 + 21/90 = 42/90 = 7/15. The routes agree, so the answer is safe.

Every pair of branches from one point adds to 1: 6/9 + 3/9 = 1 and 7/9 + 2/9 = 1. If that ever fails, fix the tree before you multiply anything. The math solver is quicker than hunting for the slip by eye.

The complement rule: how "at least one" questions are actually done

P(not A) = 1 - P(A). This turns a horrible question into a one-line calculation, because the opposite of "at least one" is always "none".

A fair die is rolled three times. Find the probability of at least one six. Directly, that means adding three separate cases: exactly one six, exactly two, exactly three. By complement: P(no six on one roll) = 5/6, so P(no six at all) = 5/6 × 5/6 × 5/6 = 125/216. Therefore P(at least one six) = 1 - 125/216 = 91/216, about 0.421.

One line instead of three cases. If you see "at least one" and you are not writing 1 minus something, you are doing more work than the mark scheme expects.

Conditional probability without the scary notation

Conditional probability just means the sample space has shrunk because you already know something. The formula is P(B given A) = P(A and B) divided by P(A).

Back to my class of 30. Given a student takes history, what is the probability they also take geography? You are no longer choosing from 30 people, only from the 18 who take history. Of those, 8 take geography, so 8/18 = 4/9.

Compare that with the plain probability of taking geography, 14/30 = 7/15. Different numbers, so the two subjects are not independent in that class. That is the reasoning AP and A-Level questions want: compare P(B given A) with P(B) and say whether they are equal.

Where students actually lose the marks

• Adding when the question says both. My mock mistake, and the most common one by a distance.
• Forgetting the denominator drops without replacement, so writing 7/10 × 6/10.
• Missing the overlap term in the addition rule and producing an answer over 1.
• Cancelling too early and losing the common denominator you need for the adding step. Keep everything over 90, then simplify once.
• Not showing the multiplication. Method marks go to correct products even when the arithmetic is wrong, so write 7/10 × 6/9 down before evaluating.
• Rounding to 0.42 when the question asked for an exact fraction.

Test yourself

1. A bag has 5 green and 4 yellow balls. Two are drawn without replacement. What is the probability both are green?
2. A coin is flipped four times. What is the probability of at least one head?
3. In a group of 40 students, 22 play football, 15 play cricket, 6 play both. What is the probability a random student plays neither?

Answers: 5/9 × 4/8 = 20/72 = 5/18; 1 - (1/2)^4 = 15/16; 1 - (22 + 15 - 6)/40 = 9/40. For more with instant marking, generate a set on the quiz tool or use the maths hub.

FAQ

How do I know if events are independent?

Ask whether the first event changes the second. Coin flips and dice rolls are independent because the object has no memory. Drawing cards or counters without replacement is dependent, because the bag has changed. Formally, events are independent when P(B given A) equals P(B).

Do I always have to draw a tree diagram?

Draw one for anything with two or more stages. It takes about thirty seconds and prevents both big errors at once, because the structure forces you to multiply along and add down. For a single-stage question a tree is a waste of time.

What is the difference between mutually exclusive and independent?

Mutually exclusive means the two events cannot both happen, so P(A and B) = 0. Independent means one happening does not change the chance of the other. They are not the same thing, and confusing them is a classic trap in CBSE and GCSE questions.

Why does my answer sometimes come out above 1?

Almost always because you added overlapping events without subtracting the overlap, or added when you should have multiplied. Any probability above 1 is impossible, so treat it as a signal to recheck the structure of your working. If you cannot spot it, paste your working into the chat tutor and ask it to check the step rather than hand you the answer.

In short

Probability is three rules and one decision. Decide AND or OR first, then multiply along or add across, subtract the overlap when events can happen together, and reach for 1 minus the opposite whenever you see "at least one". Get that decision right at the top and the arithmetic underneath almost takes care of itself.