Mathematics

Tree diagram

Two flips, two draws, two spins: how to compute probabilities across several stages cleanly.

What you need first

Flipping a single coin is easy. But what if you flip twice and want heads both times? As soon as a random experiment has several stages, it quickly gets confusing. The tree diagram brings order to it and turns a tricky question into a clear calculation.

Multi-stage random experiments

A multi-stage random experiment is when several random events happen one after another: flipping a coin twice, drawing two marbles in turn, spinning a wheel several times. Each stage has its own possible outcomes, and you want to know how likely a certain combination is at the end.

Marble bag

Blue3

Yellow5

P(Blue) = 3/8 = 37.5%

Try it: imagine drawing a marble twice in a row. Each draw is one stage of your tree diagram.

The tree diagram

In a tree diagram you draw branches for each stage, one per possible outcome. On every branch you write its probability. From the second stage on, that means the probability which applies once you have already got that far. If you draw with replacement, the numbers stay the same at every stage. If you draw without replacement, one marble is missing the second time and the numbers change. A complete route from the root to an end is called a path and describes exactly one possible overall result, for example heads and then heads again.

The path rule

You get the probability of a whole path by multiplying the probabilities along the path. For heads twice you compute $0.5 \cdot 0.5 = 0.25$ . This is the , and it is the most important trick for multi-stage experiments.

P(\text{Pfad}) = p_1 \cdot p_2 \cdot \ldots

Path rule: probabilities along a path are multiplied.

Combining several paths

Sometimes several paths lead to the same event. Exactly one heads in two flips is reached via heads then tails or tails then heads. Then you first work out each path with the path rule and add the results: $0.25 + 0.25 = 0.5$ . Multiplying goes along one path, adding goes across different paths.

Exercises

0 of 6 solved

Time to try it yourself. You can't break anything, every attempt counts.

A coin is flipped twice. What is the probability of heads twice? Compute 0.5 times 0.5.

How do you compute the probability of a single path?

A wheel shows the colour red with probability 0.2. Spin twice: how likely is red both times?

Put the steps in the right order to find the probability of an event in a multi-stage experiment.

1Multiply the probabilities along each single path.
2Add the results of the paths that lead to the event.
3Draw the branches for each stage and write its probability on each one.

A bag holds 2 red and 3 blue marbles. You draw twice and put the marble back each time, so the probability of red is 0.4. How likely is red twice?

Adding up the probabilities of all paths in a tree diagram must always give ….

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