6. Using a Probability Tree

 
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Overview

Probability trees can be used to visualize a probability problem with multiple sequential components. Learn how to create one of these trees and use it to calculate probabilities in this lesson.

Summary

  1. Lesson Goal (00:13)

    The goal of this lesson is to learn how to visualize probability problems with a probability tree.

  2. Understanding the Problem (00:23)

    We consider the example of a phone company. They need to decide whether a cheap phone model or an expensive model will be the most profitable to develop. The probability of either model being preferable varies based on whether the market expands or contracts in the next year. 

    We can estimate the probabilities of the market expanding or contracting, as well as the probabilities that the expensive phone or the cheap phone will be the most profitable in either situation. Using this information, we want to find the probability that the expensive phone will be most profitable. We can do this by drawing a probability tree, which is a visual representation of all the possible outcomes to a probability problem.

  3. Drawing the Tree (02:20)

    A probability tree consists of a series of nodes and branches, representing processes, and the possible results of those processes. In our tree, the first event is whether the market expands or contracts. There are two branches from this node, representing the two possible events. Each branch includes the probability of its event.

    The second level of the tree represents which phone is best to produce. For this level, there are two branches for each of the nodes at the end of the first level, meaning there are four branches in total. As before, we add a probability to each of these branches. Each of these is a conditional probability, as it is dependent on the outcome of the first level of the tree.

    We can add as many levels to the tree as we want. When the tree is complete, the nodes at the end of the tree are called leaves, and they represent all the possible outcomes of our problem.

  4. Calculating Final Probabilities (03:53)

    Our objective is to calculate the probability of each leaf of the tree. To calculate a leaf’s probability, we multiply the probabilities and conditional probabilities for the sequence of branches leading to that leaf.

    We should find that the probability for all the leaves adds up to 1. If we want to find the probability of some event, we simply add the probability for all the leaves where the event occurs. For example, to find the probability that the expensive phone is most profitable, we add the probability for each of the leaves where this is the case.

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