How to draw a tree diagram in probability with solved examples
Root Node
The root node has no superior parent. It itself is the parent node.
Nodes
Nodes are the family members of the root node linked with the help of branches and express the relationship among the members.
Branches
The nodes which are linked together to represent relationships and connections among them with the help of lines joining the nodes are called branches.
Leaf Nodes
Leaf nodes are also called child nodes or end nodes because they are the last members of a tree diagram. They have no further child node or children.
Where are Tree Diagram Examples are Used:
To show family relations.
In mathematics.
In the computer science field.
In taxonomy.
In business organizations.
In science of classification.
Generally, a tree diagram is used in the classification of things or to represent a sequence of events. A tree diagram starts with a parent node and branches into two or more nodes. Each node will further branch into more nodes and so on. This begins to resemble the structure of a tree. In mathematics, tree diagrams are used in statistics and probability.
Tree Diagram in Probability
In probability theory, a probability tree diagram shows all the possible outcomes. The first event is represented by a dot. Branches emerging from this dot represent all the possible outcomes. The probability of each outcome is written on its branch until a conclusion is reached.
The probability of any event tells us how likely to happen something is. In other words, it tells us about the possibility of an event taking place. If a coin is tossed, what is the sure outcome of this event?
There are two favorable outcomes, one is head and the other is tail. So the best to know how likely they are to occur is by using the probability theory.
The probability value is a numerical value and it always lies between 0 and 1. The probability of an impossible event is zero and the probability of the sure event is 1.
The formula of probability is given below
Probability of an event, P(E) = \[\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\].
Regarding a tree diagram, kindly take the following into consideration:
The tree diagram should include all the possibilities of an outcome
The sum of all the possibilities should be 1.
The number of branches shown in the tree diagram actually represents the number of different possibilities
Probabilities have to be represented by writing them on the branches of the tree.
Calculating Overall Probability and Probability Tree Diagram
Suppose we toss a coin two times, what will be the sample space?
If we toss a coin twice, the favorable outcomes will be ahead or a tail. The possible outcomes may be
Heads in both the attempts= (H,H)=HH
Head in the first attempt and Tail in the second attempt=(H,T)=HT
Tail in the first attempt and head in the second attempt=(T,H)=TH
Tail in both the attempts=(T,T)=TT
So the sample space of the event will be {HH,HT,TH,TT}.
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Steps to Calculate Overall Probability
Write the probability value on the branches. Multiply the probability value along the branches and add the probability value obtained after the multiplication.
When we add all the probability values, the result should be equal to 1.
Tree Diagram Example:
Question 1: A bag contains 3 black and 2 white balls. George picks a ball at random from the bag and puts it back in the bag. He mixes the balls in the bag and picks another ball at random from the same bag.
1- Construct a probability tree diagram.
2- Using the tree diagram, calculate the probability of picking two black balls.
Solution
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Check that the probabilities in the last column add up to 1.
2) To find the probability of getting two black balls, first locate the B branch and then follow the second B branch. Since these are independent events we can multiply the probability of each branch.
So, the probability of getting two black balls= ⅗ × ⅗ = 9/25
Questions for Practice:
From an ordinary deck of 52 playing cards, a card is picked. Then, without replacement, another card is picked. Draw a probability tree for the above-mentioned situation and using it, try to calculate the probability of picking two black cards.
Joy has been assigned some work in Science and English. His probability of completing Science work on any given day is 1/3. The probability that he will complete his English work is ¼. Both of these are independent events.
By drawing a tree diagram, calculate the following:
The probability of Joy completing his work in both the subjects
The probability that Joy will complete the homework of only one subject on a given day.
FAQs on Tree Diagram in Maths for Probability Problems
1. What is a tree diagram in maths?
A tree diagram in maths is a visual diagram used to show all possible outcomes of an event in probability. It starts from one point (the trunk) and branches out to represent different choices or results. Tree diagrams are commonly used in probability to calculate the total number of outcomes and to find the probability of combined events. Each branch represents a possible outcome, and probabilities are written along the branches.
2. How do you draw a tree diagram for probability?
To draw a tree diagram for probability, list all possible outcomes step by step using branches. Follow these steps:
- Start with a single point.
- Draw branches for the first event and label them with outcomes and probabilities.
- From each branch, draw further branches for the next event.
- Continue until all stages are shown.
- Multiply probabilities along each path to find combined probabilities.
3. How do you calculate probability using a tree diagram?
To calculate probability using a tree diagram, multiply along the branches and add across final outcomes when necessary.
- Multiply probabilities on the same path to find joint probability.
- Add probabilities of different paths if they lead to the same final event.
4. What is the formula used in a tree diagram?
The key formula used in a tree diagram is P(A and B) = P(A) × P(B|A). This is called the multiplication rule of probability. For independent events, it simplifies to P(A and B) = P(A) × P(B). This formula is applied along each branch of the tree to find combined probabilities.
5. Can you give an example of a tree diagram in probability?
A simple example of a tree diagram is tossing a coin and then rolling a die.
- First event (coin): H (1/2), T (1/2).
- Second event (die): 1–6 (each 1/6).
6. What is the difference between a tree diagram and a sample space?
A tree diagram visually shows outcomes step by step, while a sample space is the complete list of all possible outcomes. The tree diagram organises outcomes using branches, making multi-step events easier to follow. A sample space may simply list outcomes like {HH, HT, TH, TT} without showing stages visually.
7. When should you use a tree diagram?
A tree diagram should be used when dealing with multi-step probability events or dependent and independent events. It is especially helpful when:
- Events happen in sequence.
- You need to avoid missing outcomes.
- Probabilities change after each event (dependent events).
8. How do tree diagrams work for dependent events?
For dependent events, tree diagrams use conditional probabilities on the second set of branches. The probability of later outcomes changes based on earlier results. For example, drawing two balls without replacement from a bag of 3 red and 2 blue balls: Probability of Red then Red = 3/5 × 2/4 = 6/20 = 3/10. The second probability changes because one red ball was removed.
9. What are common mistakes when using tree diagrams?
Common mistakes in tree diagrams include not multiplying along branches correctly and forgetting to adjust probabilities for dependent events. Other errors include:
- Missing possible outcomes.
- Adding probabilities instead of multiplying along a path.
- Incorrect total probability not adding up to 1.
10. Do tree diagram probabilities always add up to 1?
Yes, the total probability of all final outcomes in a tree diagram must equal 1. Since the tree represents all possible outcomes of an experiment, their combined probabilities must sum to 1. If they do not, there is likely a calculation or branching error.
