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 1^{st} toss and the 2^{nd} toss.
The tree diagram below represents tossing a coin twice in a row:
In the 1^{st} toss, the coin can land Heads (H) or Tails (T). This is seen in the left-most branches.
In the 2^{nd} toss, the coin can also land Heads (H) or Tails (T). This is seen in the branches that come from each branch from the 1^{st} 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
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 1^{st} coin toss, and a Head in the 2^{nd} 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
showNote
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: