Probability Tree Diagram

A probability tree diagram is a visual tool that shows all possible outcomes of an event in a clear, step-by-step format. It includes a starting point, branches, nodes, probability labels, and endpoints. Each branch represents a possible event, and each endpoint means an outcome.
 

By multiplying the probabilities along each branch and adding the probabilities of the required paths, probability trees make it easier to solve complex probability problems. They are commonly used in areas like decision-making, risk assessment, and statistical analysis.

For example,
If you toss a coin, the probability tree will show two branches: one for getting heads and one for getting tails, each with a probability of ½.

There are essential parts that come under the probability tree, and to understand the concept of the probability tree diagram, we must know the parts of the probability tree diagram:

 Roots: The diagram starts at the root, which represents the start of the experiment. All first events branch out from this point.

 
 Branches: Branches are the lines extending from a node that show the different possible outcomes of an event. Each branch is labeled with its outcome and probability.


Nodes: Nodes are points where branches split into further outcomes. A node can lead to more branches or represent an outcome.


Probability labels: Each branch carries a probability label representing the likelihood of that outcome. The probabilities from a single node must always add up to 1.


Paths: A path is the sequence of branches from the root to an outcome. The probability of a path is the product of the probabilities along it.


 End points: End points, or terminal nodes, show the final results of the experiment. Their probabilities are obtained by multiplying all the probabilities along the path.


Summation of probabilities: If an outcome can happen in more than one way, add the probabilities of all paths leading to it.

Given below are the steps that are used to find the probability using a tree diagram:

• Step 1: Define the experiment: Students are supposed to identify the type of event or experiment it is. They have to determine whether the event is dependent or independent.

• Step 2: Draw the probability tree diagram: Start with a single point or node and draw branches for each possible outcome for each possible outcome of the first event. Label the branches with the event name and the probability of it occurring. If there is more than one stage in the event; extend the branches for each possible outcome of the second event from each first event branch. Repeat this process for all stages.

• Step 3: Assign probabilities for each branch: Ensure that when you add up all the probabilities, the result is 1. If the event is independent the probability remains the same (as in flipping a coin, where both are independent outcomes), and if the event is dependent the probability changes based on the previous outcome (like picking cards from a deck without putting them back).

• Step 4: Multiply the probabilities along the path: The probabilities along the corresponding path in the tree are multiplied to find the probability of a particular group of events.  For Example, the tree given below has the probability of two coins tossed:

We can draw a probability tree with the help of the following steps:

• Step 1: Identify the experiment and events: The students must clearly define the events, and determine whether the events are independent or dependent. Students must start with a single point, then draw branches for each possible outcome of the first event. They must label the branch with the event name and the probability of that event occurring.

• Step 2: Draw the first branches for the initial event: Students must start with a single point, then draw branches for each possible outcome of the first event. They must label the branch with the event name and the probability of that event occurring.
   

• Step 3: Extend branches for subsequent event: If there is a second stage in the experiment, extend new branches from each first event outcome. Label the branches with the event name at that stage and the probability of that event considering the dependencies.

• Step 4: Check probabilities at each branching point: Students must ensure that the sum of probabilities at each branch level is 1, they must also know that if the event is independent then the probability remains the same. If the events are dependent, the students must update the probabilities accordingly.

• Step 5: Multiply the probabilities along the path: To find the probability of a specific event and their outcomes, multiply the probabilities along the path.

• Step 6: Sum probabilities for desired outcomes: If the students need to find the probability of an event that can occur in multiple ways, sum the probabilities of all relevant paths.

• Step 7: Verify the diagram and probabilities: Students must double-check that all probabilities are correct and sum to 1 at each stage, ensure all the possible outcomes are included in the tree, and confirm that the branches are labelled correctly and calculations are clear.

A conditional probability tree diagram is a helpful tool for showing the outcomes of dependent events, events in which the result of one event affects the result of the next. It visually represents how earlier outcomes change the probabilities of later ones.

For example, a person wants to know the probability of being late to work, depending on the weather. The likelihood of rain is 0.4. If it rains, the probability of being late is 0.7. If it does not rain (probability 0.6), the possibility of being late reduces to 0.2.

Here, we want to find the probability that the person is not late on a rainy day.

Step 1: Start the tree with the weather.

Where, Rain = 0.4 and No Rain = 0.6


Step 2: For the "Rain" branch, draw two outcomes: late = 0.7 and not late = 0.3 because \( 1 – 0.7 = 0.3\). 


Step 3: For the "No Rain" branch now, draw two outcomes:

Late = 0.2

Not late = 0.8


Step 4: Multiply along the branch for,
 
Rain → Not Late

Now, the probability = \(0.4 × 0.3 = 0.12\)

Understanding probability tree diagrams can be challenging, so using a few helpful strategies can make the topic easier to learn. Below are some tips and tricks that will help you master probability trees:
 

• Practice drawing trees for basic situations, such as coin tosses or dice rolls, before moving on to more complicated problems.
  
   
• Make sure each branch shows the event and its probability, so the diagram stays straightforward to read.
  
   
• To find the probability of a sequence of events, multiply the probabilities as you move along the path.
  
   
• If an outcome can happen in more than one way, add the probabilities of all paths that lead to that outcome.
  
   
• The combined probabilities of all outcomes in the tree should always add up to 1.
  
   
• Parents and teachers can help children begin by drawing trees for basic events, such as tossing a coin or rolling a die, before moving on to more complex problems.
  
   
• Children should label each branch with the event and its probability, while teachers can guide them on organizing the diagram correctly.
  
   
• Teachers and parents can encourage children to double-check that the sum of all outcomes is always 1.

There are a lot of real-life applications of probability tree diagrams. Let us now see the applications and uses of probability tree diagrams in our day-to-day applications:

• Business and decision-making: We use probability tree diagrams in business and decision-making as it helps investors to analyze different investment outcomes based on the market conditions, factories use it to estimate how likely a product is to pass or fail quality checks.
   

• Medicine and healthcare: We use probability tree diagrams in medicine and healthcare as it helps doctors to calculate the probability of a patient having a disease based on test results, pharmaceutical companies assess the effectiveness of drugs under different conditions.
   

• Engineering and manufacturing: We use probability tree diagrams in engineering and manufacturing, as it helps engineers to predict how often a machine will fail based on maintenance schedules, car manufacturers assess the probability of a car part failing based on warranty period. 
  
  
   
• Education and examinations: In education, teachers and examiners use probability tree diagrams to analyze students’ performance.
  
  
   
• Finance and banking: In finance, probability tree diagrams are used to evaluate risks and returns.