Probability Tree Diagram

Probability tree diagram visually represents all possible outcomes of an event. It is a step-by-step representation of different events and probable outcomes of those events. The probability tree consists of a starting point, branches, nodes, probability labels and the end points.

By multiplying the probabilities along the branches and summing relevant paths, probability tree diagrams help in solving complex problems. They are widely used in decision-making, risk analysis, and statistical calculations.
 

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

• Roots (Starting Point): The starting point of the probability tree diagram is known as the Roots. All the first possible events will start from the Roots.

• Branches: These are the lines that extend from a node to represent different outcomes of an event. Each branch is labeled with the name of the outcome and the probability of that outcome.

• Nodes (Decision Points): These are the points where the branches split into different possible outcomes. A node can start new branches and represent final outcomes.

• Probability Labels: Each branch represents a possible outcome of an event, which is called as probability label. When you add up the probabilities of one branch, they must always equal to 1. 

• Paths: Paths are sequences of branches from the root to a final outcome. It represents a specific outcome sequence in the experiment. The probability of a path is found by multiplying the probabilities along the branches.

• End Points (Final Outcomes): The end points, also called the terminal nodes of the tree, represent all the final results of the probability experiment. Each end point has a total probability that is calculated by multiplying along the path.

• Summation of Probabilities: If a probability question involves multiple ways to get the same result, the probabilities of different paths leading to the same outcome must be added together. 

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:

• 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 used to show all the possible outcomes of a dependent event. Let us understand the conditional probability tree diagram with the following example:

If a student attends two tests, the probability of clearing the first test is 0.7. If the first test is cleared, then the probability of clearing the second test is 0.8. If the student fails the first test, the probability of clearing the second test is 0.6. This example is illustrated in the following diagram. 

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.