There are different types of probabilities that can help us determine the likelihood of an event happening.
Identifying the right type of probability to use is an important first step — it determines what information you actually need, and whether you're pulling it from the right place.
Let's use an example to illustrate this.
You're waiting in line at airport security when you notice TSA pull a bag off the belt and flag it.
You wonder: what is the likelihood that a bag flagged by TSA is actually dangerous?
Let's work it out using a probability table.
We'll use these facts to build a probability table.
Rows: whether a bag is actually dangerous or safe.
Columns: whether it was flagged or not.
Interior cells: the cells in the middle of the table represent bags that have both characteristics.
Total: the edge cells represent the sum of that row or column.
| Flagged | Not flagged | Total | |
|---|---|---|---|
| Dangerous | — | — | — |
| Safe | — | — | — |
| Total | — | — | 1,000 |
Rows: whether a bag is actually dangerous or safe.
Columns: whether it was flagged or not.
Interior cells: the cells in the middle of the table represent bags that have both characteristics.
Total: the edge cells represent the sum of that row or column.
Hover, tap, or tab into a cell for a hint. Numbers aren't checked until you press Check.
| Flagged | Not flagged | Total | |
|---|---|---|---|
| Dangerous | |||
| Safe | |||
| Total | 1,000 |
This occurs at the margins of the table — a row or column total, divided by the grand total.
We can write this in probability notation as follows:
What is the probability that a bag is dangerous?
This occurs in a single interior cell of the table — the probability that two events both happen.
We can write this in probability notation as follows:
Order doesn't matter here: P(Flagged and Dangerous) and P(Dangerous and Flagged) are the same cell, so the same number. Hold onto that — on the next page you'll meet a probability where order changes the answer completely.
Joint probability is useful when you need to know how often two events occur together — for example, estimating how many bags will be both flagged and turn out to actually be dangerous, so a screening team can staff and budget accordingly.
What is the probability that a bag is flagged and dangerous?
This doesn't sit directly in the table — you derive it from two numbers already there, once you know one event happened.
We write this in probability notation as follows:
What this means: in the universe of flagged bags, what is the probability that a bag is dangerous?
Unlike a joint probability, order matters. P(Flagged | Dangerous) asks something entirely different: of the bags that are dangerous, how many get flagged?
Conditional probability is often what you actually want to know — it tells you the probability of an event happening, given that another event has already happened. Conditional probability shrinks the universe of possibilities: marginal and joint probabilities consider all 1,000 bags, whereas conditional probability considers only the outcomes where the conditioning event — the one after the bar — has already occurred. For example, conditional probability can tell you:
What is the probability a bag is dangerous, given that it was flagged?
Now write this in probability notation:
That last one is new — but you can read it straight off the same table: of the 3 dangerous bags, 2 were flagged. 2 ÷ 3 = 0.667 = 66.7%.
Look at the last two rows. They use the same two events and the same three numbers, and they differ by a factor of more than thirty: 2% versus 67%. Flipping what's in front of the bar is not a cosmetic change — it asks a different question and gets a different answer.
Both are true at once, and each answers a question someone actually cares about. The screening team wants P(Flagged | Dangerous): are we catching the threats? Two out of three — better than nothing, worse than you'd hope. The traveler pulled out of line wants P(Dangerous | Flagged): does this flag mean anything about me? Almost never — 98 of every 100 flagged bags are perfectly safe. Treating the reassuring 67% as though it settled the traveler's question is the error, and it has a name: confusing the inverse. It shows up constantly in policy work — in medical screening, in predictive risk scores, in courtroom statistics.
Now compare a different pair: P(Dangerous) = 0.3% and P(Dangerous | Flagged) = 2%. Both are small, but they are not the same number. Before you knew anything about a particular bag, your best estimate of danger was 0.3%. The moment you learned it had been flagged, that estimate jumped nearly sevenfold, to 2%. The flag is informative — it just doesn't make the bag likely to be dangerous.
Updating a probability in light of new evidence — and keeping straight which conditional you're actually holding — is exactly what policymakers have to do constantly. It also has a formal tool behind it: Bayes' rule, which is the machinery for turning one of those conditionals into the other.