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6. Using a Probability Tree
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.
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Lesson Goal (00:13)
The goal of this lesson is to learn how to visualize probability problems with a probability tree.
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.
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.
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.
In the previous lessons, we've seen a lot of math and formulas.
We're now going to look at a more visual approach to probability.
In this lesson, we'll learn how to visualize probability problems with a probability tree. We'll consider the example of a phone company.
They're planning on launching a new model this year, and have to decide what kind of product to develop. They can choose a cheap model with basic specifications and a low price point, or an expensive model with advanced features at a high price. They want to know which model will be the most profitable. The best phone to produce depends on the growth of the economy and the market for phones in the next year. If the market grows, the expensive phone is likely to be the most profitable, with a probability of 0.8.
If the market contracts, the cheap phone is likely to be the most profitable, with a probability of 0.7.
The company believes that the probability of the market growing is 0.7.
Given all this information, our objective is to identify the probability that the expensive phone will be the most profitable. Let's consider this information in terms of events.
We'll say that M+ is the event where the market grows and M- is the event where the market contracts.
E is the event where the expensive phone is most profitable, and C is the event where the cheap phone is most profitable. We know the probability of M+ is 0.7, and the probability of M- is 0.3.
The probability of E given M+ is 0.8, and the probability of C given M+ is 0.2.
Finally, the probability of E given M- is 0.3, and the probability of C given M- is 0.7.
Our objective is to find the probability of E.
We'll represent this problem using a probability tree.
This is a visual representation of all the possible outcomes for a probability problem. Let's start drawing our tree. First, we'll consider if the market grows or contracts. We'll add two branches, one for the market expansion event, M+, and one for the market contraction event, M-.
We'll also add the probabilities for these events to the branches. Next, we'll add a second level to the tree.
Let's consider the scenario where the market expands.
There are two possible events here.
Either the expensive phone will be most profitable, or the cheap phone will be most profitable.
We'll add a branch for each of these possibilities. because we're only considering the situation where the market expands, the events at the ends of these branches will be conditional events.
We'll also add the probabilities for these events, which are conditional probabilities that we determined previously.
Now let's consider the case where the market contracts.
Again, there are two possible events here, so we'll add two branches.
As before, these events are conditional on the market contracting, and the probabilities are conditional probabilities.
Our tree is now complete.
The four events at the right of the tree are called the leaves of the tree.
They represent all the possible outcomes of our problem.
Our next task is to calculate the probability for each of these leaves.
To do this, we simply multiply the probabilities for each branch that we cross to reach that leaf.
The first leaf represents the situation where the market expands and the expensive phone is most profitable.
The probability of reaching this leaf is the probability of the market expanding times the conditional probability of the expensive phone being the most profitable, given the market expanding. As we can see from the leaves, this is 0.7 times 0.8, which is 0.56.
The second leaf represents a situation where the market expands and the cheap phone is most profitable.
To calculate the probability of reaching this leaf, we multiply 0.7 by 0.2, which is 0.14.
We can also calculate the probability for the two leaves where the market contracts in the same way.
We find that the probability of the market contracting and the expensive phone being most profitable is 0.09.
The probability of the market contracting and the cheap phone being the most profitable is 0.21.
We now have the probability for each leaf.
We should find that the probability values for all leaves add up to one, because the leaves represent every possible outcome for the problem.
This is the case with our tree.
Our aim is to calculate the probability of E. That is, the probability that the expensive phone will be the most profitable.
To do this, we need to add the probability for each of the leaves where the expensive phone is most profitable. There are two leaves where this is the case. So the probability is 0.56 plus 0.09, which is 0.65.
We can also calculate the probability of the cheap phone being the most profitable by adding the values for the other two leaves.
This is 0.14 plus 0.21, which is 0.35.
This tells us that there is a probability of 0.65 that the more expensive phone will be the most profitable. Given all the information we have about the problem, the probability of the cheap phone being most profitable is 0.35. Of course, changing any of the branch probabilities would change their probabilities for some or all of the leaves. Note that changing any of the branch probabilities would change the probabilities for some or all of the leaves, so it's important that we're confident of the inputs before we create a probability tree. Let's stop the lesson here. In the next lesson, we'll move on to a new topic and learn about permutations.