Probability tree

In a probability tree you multiply along a single path to get the probability of that whole sequence. If an event can happen along more than one path, you add those paths together. Every path in the tree adds up to exactly 1, because something has to happen.

Along a path you multiply

Each branch carries the probability of one step. Following a path from left to right means the steps happen one after another, and the probability of the whole run is the product of the branches you pass.

Set the probability to 0.50 and every path in a two-stage tree comes to 0.25, which is right: there are four equally likely orders. Push it to 0.80 and the four paths spread far apart, while their total stays put.

Across paths you add

A question like "exactly one hit" does not point at one path but at several. Two stages give two of them, three stages give three. You work each one out and add the answers together.

Switch that layer on and the paths involved light up. This is the split that goes wrong most often in tests: multiply along a path, add across paths.

At least once: work out the opposite

For "at least one hit" over three stages you would have to add seven paths. Only one path is the case where it never happens, so work out the probability of no hits at all and take it away from 1.

That is the complement rule. The more stages you add, the bigger the difference in effort, and the more likely it is that the exam wanted the short way.

Frequently asked questions

How do you draw a probability tree?

Start from a single point, draw one branch per possible outcome and write the probability on each branch. Repeat for the next step. The right-hand side ends with one path per possible order, and its probability is the product of the branches leading to it.

Why do you multiply along a path?

Because each step keeps only a share of what is left. With a probability of 0.5 you keep half the cases, then half of that half, so a quarter overall. That holds as long as the steps do not affect one another.

When do you add probabilities instead of multiplying?

As soon as an event can happen along more than one path. Multiplying goes with "and then", adding goes with "or". Two hits is a single path; exactly one hit is several.

How do you work out the probability of at least one?

Work out the probability that it happens zero times and take it away from 1. With three attempts at 0.3 that is 1 minus 0.7 cubed, about 0.657.

Does this tree also work without replacement?

No. This tool repeats the same attempt, so the probability on every branch stays the same. If you draw marbles without putting them back, it changes after each draw and every branch carries a different value. Multiplying along the path still works either way.

More interactive maths tools

• Binomial distribution →Set n, p and k to see the bars and the formula, with the normal approximation alongside.
• Boxplot and spread →Type in your own numbers and watch the boxplot, the quartiles and both standard deviations move live.
• Normal distribution →Drag the bounds across the bell curve and read probability and z-scores straight off.