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How to Find Probabilities of Each Outcome from a Tree Diagram (Mathematics Lesson)

How to Find Probabilities of Each Outcome from a Tree Diagram - a Simple Case

The tree diagram below represents tossing a coin.

When the tree diagram represents a single event, the probability of each outcome is simply the probability written by each branch.

The 'And' Rule of Probability from a Tree Diagram

The above example was very simple, as it looked at the tree diagram for a single event.

But we might be interested in finding out the probability that one event and another event occurs. In this case, we use the 'and' rule of probability.

For example, a coin can be tossed twice. We can work out the probability of the result of the 1st toss and the 2nd toss.

The tree diagram below represents tossing a coin twice in a row:



In the 1st toss, the coin can land Heads (H) or Tails (T). This is seen in the left-most branches.

In the 2nd toss, the coin can also land Heads (H) or Tails (T). This is seen in the branches that come from each branch from the 1st toss.

The four possible outcomes of the two coin tosses are listed at the end of each final branch:

  • Head, Head

  • Head, Tail

  • Tail, Head

  • Tail, Tail
It is then possible to calculate probabilities.

Multiply Along the Branches to Find the Probability of Each Outcome

To find the probability of each outcome ('Head, Head', 'Head, Tail' etc. ) multiply along the branches.

Question: What is the probability of getting a Head in the 1st coin toss, and a Head in the 2nd coin toss?

Multiply the probabilities along the branches:





The probability of getting Head, Head is ¼.

By a similar process, it can be shown that the probability of each outcome is ¼.



Note: Adding up the overall probabilities from each outcome gives 1, as always should be the case:


A Real Example of How to Find the Probabilities of Each Outcome from a Tree Diagram

In the above example of two coin tosses, each event was independent.

The slider below gives a real example of using a tree diagram to calculate the probabilities from a dependent event.
Interactive Test
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Note
WHAT IS PROBABILITY?

Probability tells us how likely something is to happen.

The probability of an event is defined as:



Probability is given as a number between 0 and 1.

A probability of 0 means an event is impossible.

A probability of 1 means an event is certain.

The closer a probability is to 0 the less likely it is. The closer a probability is to 1 the more likely it is.



WHAT IS A TREE DIAGRAM?

A tree diagram displays the possible outcomes from an event, and allows the probabilities of different outcomes to be calculated.

SIZE OF TREE DIAGRAMS

Tree diagrams can be any size.

It would be possible to draw a tree diagram for three coin tosses:



Top Tip

A USEFUL CHECK

There is a useful way to check that the probabilities on a tree diagram are all right.

Firstly, notice that at each branch, the probabilities add up to 1:



Secondly, when the probabilities of each final outcome are known, they also add up to 1: