When do you add probabilities and when do you multiply?

Add when you want one event or another on a single occasion, and multiply when you want one event and another across separate stages. Adding requires the events to be mutually exclusive, or you must subtract the overlap you counted twice. Multiplying requires the second probability to be the one that applies given the first has happened, which is why without-replacement problems change the second fraction.

What is the easiest way to find the probability of at least one?

Find the probability of none and subtract it from one. At least one covers many separate cases — exactly one, exactly two, and so on — while its opposite is a single case that multiplies straight down one branch of the tree. For three dice, the chance of at least one six is one minus five sixths cubed, which is about 0.42. The complement is not a shortcut, it is the standard method.

What does conditional probability actually mean?

It is the probability of one event given that another has already happened, and it means the sample space has shrunk. P(A given B) is worked out inside B alone, so the denominator becomes the number of outcomes in B rather than the total. Written as a formula it is P(A and B) divided by P(B). On a two-way table it is reading down one column instead of across the whole table.

Why should you switch doors in the Monty Hall problem?

Because your first choice was right one time in three, and that has not changed. The host always opens a losing door, so his action gives no information about your door but concentrates all the remaining probability onto the one door he did not open. Staying wins one third of the time, switching wins two thirds. The key is that the host knows where the prize is and never opens it by accident.

How can a positive test result still mean you are probably healthy?

Because when a condition is rare, the false positives from the huge healthy group outnumber the true positives from the tiny affected one. With a disease affecting one person in a thousand and a test that is 99 percent accurate, testing 100,000 people gives about 99 true positives and about 999 false positives, so a positive result is right only about nine percent of the time. The test is accurate; the base rate is what determines the answer.

Why do only 23 people give a fifty-fifty chance of a shared birthday?

Because the question is about pairs, not people, and 23 people form 253 pairs. Each pair has a small chance of matching, but there are enough of them that the chance of no match at all falls below a half. The intuition that fails is comparing yourself to everyone else, which is only 22 comparisons; the event asks about every pair in the room.

Probability and Combined Events

Probability is the one branch of school mathematics where a correct calculation regularly contradicts a strong intuition — and where the intuition is usually the thing that is wrong. That makes it unusually worth doing carefully. The arithmetic is easy; the difficulty is always in stating precisely what event you are asking about.

What does a probability actually measure?

The share of outcomes in which the thing happens, on a scale from 0 (impossible) to 1 (certain). The Probability Scale puts everyday language onto that line — unlikely, even chance, almost certain — which is the first step to replacing the words with numbers.

Getting a number requires listing what could happen. Sample Space is the discipline of writing out every outcome, and it is the step people skip and then regret: the sample space for two dice has 36 entries, not 11, because a 3 and a 4 is a different outcome from a 4 and a 3.

Single-Event Probability then divides — favourable outcomes over total outcomes — with the condition that makes the division legitimate: the outcomes must be equally likely. Complementary Events supplies the other half of every question: the chance something does not happen is 1 minus the chance it does, and that reversal solves more problems than any other single idea on this page.

Probability from experiment rather than theory

Not every situation has countable equally likely outcomes. A drawing pin does not. Experimental Probability estimates the chance from observed frequency instead — the number of times it happened over the number of trials.

The Law of Large Numbers says that estimate converges: the more trials, the closer the relative frequency settles to the true probability.

What the law does not say is that a run of heads makes tails "due". The coin has no memory, and each toss is independent. What actually happens is dilution: after a run of five heads, the effect of that run on the running proportion shrinks as thousands more tosses accumulate around it. The imbalance is not corrected, it is outweighed — which is the whole difference between the law of large numbers and the gambler's fallacy.

Expected Counts from a Chance runs it the other way: multiply the probability by the number of trials to predict how many times something should occur. Twenty rolls of a die should give roughly three or four sixes, and the gap between that and what happened is exactly the quantity every statistical test later measures.

How do you handle two events at once?

Draw them. Naming the Regions of a Venn Diagram is the map: A only, B only, both, and neither — four regions that account for everything, and most probability confusion is a failure to say which of the four is meant.

Possibility Diagrams is the grid version for two-stage experiments, with the outcomes of one stage along the top and the other down the side. Two dice become a 6 × 6 grid where you can literally count the cells summing to 7, and Chance from Areas extends the same idea to continuous cases, where probability becomes a ratio of areas rather than a count.

Mutually Exclusive Events is the condition under which you may simply add: if two events cannot both happen, the chance of one or the other is the sum. When they can overlap, addition double-counts, and The Addition Rule for Overlapping Events subtracts the overlap once: P(A or B) = P(A) + P(B) − P(A and B).

Independent Events is the condition under which you may simply multiply: if one event's outcome does not affect the other's chance, the probability of both is the product. Two coin tosses are independent; two cards drawn without replacement are not.

How do tree diagrams work?

Each branch carries a probability, you multiply along a path to get the chance of that whole sequence, and you add the paths that satisfy the question. Tree Diagrams establishes the machinery on independent stages, where every layer's probabilities are the same.

The Chance of At Least One is the first serious payoff. "At least one six in three rolls" spreads across seven paths; "no six at all" is one path. 1 − (5/6)³ ≈ 0.42. Doing it the long way is not wrong, it is just an invitation to lose a case.

Then the branches start changing. Without Replacement is the situation where taking one item alters what is left, so the second layer's fractions differ from the first — and differ per branch. Multiplying Along a Tree Without Replacement works the full calculation.

A bag holds 5 red and 3 blue. P(both red) = 5/8 × 4/7 = 20/56. The denominator dropped because a ball is gone; the numerator dropped because the ball that went was red.

Every without-replacement tree is already conditional probability, before the phrase is introduced. That second fraction, 4/7, is P(second red given first red) — it is not a new rule but the same one, written where the shrinkage is visible. Which is why the formal definition later should feel like a name for something you were already doing.

What is conditional probability?

The probability of one event given that another has happened, which means the sample space has shrunk to just the outcomes where the given event occurred. Conditional Probability teaches that shrinkage directly, and Conditional Probability Notation writes it: P(A | B) = P(A ∩ B) / P(B).

Read that formula as a restriction rather than a division. The numerator counts outcomes where both happened; the denominator counts every outcome in the new, smaller world. On a two-way table it is reading down a single column and ignoring the rest — which is why frequency tables and conditional probability are usually taught together, and why the charts and displays guide covers the frequency tree.

Independence gets its real definition here: A and B are independent exactly when P(A | B) = P(A) — knowing B changes nothing.

Why do the famous problems catch everybody?

Because each one hides a condition that the intuitive answer ignores.

The Monty Hall Problem: three doors, one prize, you pick one, the host opens a losing door and offers a swap. Switching wins two times in three. Your original door was right one time in three and no new information touched it, so the remaining two-thirds is now concentrated on the single unopened door. The condition people miss is that the host knows where the prize is and never opens it — remove that and the puzzle dissolves into an even chance.

The Birthday Paradox: in a room of 23 people it is more likely than not that two share a birthday. The instinct compares you against 22 others; the event compares every pair, and 23 people form 253 pairs. Computing it as 1 minus the chance of all different — 365/365 × 364/365 × 363/365 × … — makes the collapse visible.

Why a Positive Test Can Still Mean Healthy is the one with consequences outside a classroom. Take a disease affecting 1 in 1,000 and a test that is 99% accurate both ways, and run 100,000 people through it: about 99 of the 100 who have it test positive, and about 999 of the 99,900 who do not also test positive. A positive result is correct about 9% of the time.

Nothing is wrong with the test. The error is swapping two conditional probabilities: P(positive | ill) really is 0.99, but the patient wants P(ill | positive), and those are different numbers whenever the two groups are different sizes. This is the same reversal that makes a p-value misread as the probability a hypothesis is false, in the sampling and inference guide.

Reversing a Tree: the Base-Rate Answer is the general method behind it. Build the tree in the order events happen, then read it backwards: the answer to "given the test was positive, what is the chance of illness" is the one relevant path divided by every path ending in a positive. That is Bayes' theorem, arrived at by counting rather than by formula.

The mistakes worth naming

Adding probabilities of events that can both happen. Subtract the overlap, or you have counted it twice.
Multiplying when the events are not independent. Without replacement, the second fraction must change.
Listing outcomes that are not equally likely. Two dice have 36 equally likely outcomes; their 11 possible totals are not equally likely.
Believing a run is "due". Independent trials have no memory. The proportion evens out by dilution, not by correction.
Swapping P(A | B) with P(B | A). This is the base-rate error, and it is the most consequential mistake in the topic.
Enumerating "at least one" case by case. Use the complement; the long way loses a branch.

Where this leads next

Counting outcomes stops being feasible once the sample space is large, and the fix is to count without listing — permutations, combinations and the set notation that organises them, in the sets, counting and distributions guide, which also turns repeated trials into the binomial distribution. Turning an observed frequency into a claim about a population is the sampling and inference guide. And the fraction arithmetic every tree diagram runs on is in the fractions guide.

Learn this properly in the app

Math Challenge teaches each of these as an illustrated lesson — a drawn tree or Venn diagram, a worked calculation and try-it problems that re-teach the exact question you missed rather than a nearby one. They sit inside a catalog of 800+ lessons running from first counting to university-entrance probability.

Your turn

Three to try — tap what you get.

A fair coin is tossed twice. P(two heads)?

One roll of a fair die. P(even)?

A bag holds 3 red and 2 blue. P(red)?

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