How Tree Diagrams Help Solve Probability Questions
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: Step-by-Step Guide
1. What is a tree diagram in Maths?
A tree diagram in Maths is a visual tool used to display all possible outcomes of a sequence of events. It is a type of graph that uses branching lines to represent each possible outcome. The 'tree' starts with a single node (the starting point) and branches out to show the different paths or choices, making it easy to see the entire sample space of an experiment.
2. How do you create a tree diagram step-by-step?
To create a tree diagram, follow these simple steps:
- Step 1: Start with a single dot or node representing the beginning of the experiment.
- Step 2: Draw one branch from the starting node for each possible outcome of the first event. Label the end of each branch with the outcome.
- Step 3: From the end of each of these branches, draw new branches for all possible outcomes of the second event.
- Step 4: Continue this process for all subsequent events in the sequence.
- Step 5: To find the total outcomes, you can count the number of branches at the very end of the diagram.
3. Can you provide a simple example of a tree diagram?
Certainly. Imagine you flip a coin twice. The tree diagram would look like this:
- From your starting point, draw two branches for the first flip: one for Heads (H) and one for Tails (T).
- From the end of the 'Heads' branch, draw two more branches for the second flip: one for Heads (HH) and one for Tails (HT).
- Similarly, from the end of the 'Tails' branch, draw two more branches: one for Heads (TH) and one for Tails (TT).
The final outcomes listed at the end of the branches are HH, HT, TH, and TT.
4. How are probabilities calculated using a tree diagram?
To calculate probabilities, you first write the probability of each outcome on its corresponding branch. To find the probability of a specific sequence of events, you multiply the probabilities along the branches that make up that sequence. For example, if the probability of getting Heads is 0.5, the probability of getting two Heads in a row (H-H) is 0.5 × 0.5 = 0.25.
5. When is a tree diagram more useful than a simple list or a table?
A tree diagram is particularly useful when dealing with a sequence of events, especially when the probabilities change at each stage (known as conditional probability or dependent events). While a list or table can show the final sample space, a tree diagram clearly illustrates the step-by-step process and the probabilities associated with each stage, making it easier to understand and calculate complex outcomes.
6. How does a tree diagram handle dependent events versus independent events?
The structure of the diagram helps clarify the difference:
- For independent events (like a coin flip), the probabilities on the branches for the second event are the same regardless of the outcome of the first event.
- For dependent events (like drawing cards without replacement), the probabilities on the branches for the second event will change based on the outcome of the first event. For example, the probability of drawing a second King from a deck changes if the first card drawn was also a King. The tree diagram visually represents these changing probabilities on the respective branches.
7. What is the importance of a tree diagram in understanding probability?
The primary importance of a tree diagram is its ability to organise and visualise all possible outcomes of a random experiment in a clear, systematic way. It breaks down a complex probability problem into smaller, more manageable steps. This is especially important for students learning about compound events, as it helps prevent them from missing any possible outcomes when calculating the total sample space and probabilities.
8. What are some real-world applications of tree diagrams?
Beyond simple coin flips, tree diagrams are used in various real-world scenarios. They can model business decisions by mapping out potential outcomes of different strategies. In genetics, they can show the possible genetic makeup of offspring. They are also used in computer science for decision-making algorithms and in sports to analyse the chances of a team winning based on a sequence of plays.
