How To Draw A Tree Diagram – Identifying probability in mathematics can be confusing, especially since there are many rules and procedures involved. Fortunately, there is a visual tool called a probability tree diagram that you can use to organize your thinking and make calculating probabilities much easier.
At first glance, a probability tree diagram can be complicated, but this page will teach you how to read a tree diagram and how to use it to calculate probabilities in a simple way. Follow step by step and you will soon become a master at reading and creating probability tree diagrams.
How To Draw A Tree Diagram
Let’s start with a common probability event: tossing a coin that has heads on one side and tails on the other:
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This simple probability tree diagram has two branches: one for each possible head or tail. Note that the result is placed at the endpoint of a branch (this is where a tree diagram ends).
Also, note that the probability of each result is written as a decimal or fraction in each branch. In this case, the probability of each of the results (flipping a currency and getting heads or tail) is
Note that this tree diagram shows two consecutive events (the first reversal and the second reversal), so there is a second set of branches.
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Using the tree diagram, you can see that there are four possible outcomes when you toss a coin twice: heads/heads, heads/tails, tails/heads, tails/tails.
And since there are four possible outcomes, there is a 0.25 (or ¼) probability of each outcome occurring. So, for example, there is a 0.25 chance of getting heads twice in a row.
The rule for finding the probability of a particular event in a probability tree diagram is to multiply the probabilities of the corresponding branches.
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For example, to prove that there is a 0.25 probability of getting two heads in a row, multiply 0.5 by 0.5 (because the probability of getting heads on the first flip is 0.5 and the probability of getting heads in the second flip is also 0.5).
Repeat this process for the other three outcomes as follows, then add all the outcome probabilities together as follows:
· The probability of getting at least one tail from two consecutive spins is 0.25 + 0.25 + 0.25 = 0.75
What’s Going On In This Graph?
Note that in the coin toss tree diagram example, the outcome of each coin toss is independent of the outcome of the previous toss. That is to say, the result of the first toss has no effect on the probability of the result of the second toss. This condition is known as an independent event.
Unlike an independent event, a dependent event is an outcome that depends on an event that happened before. These types of situations are a bit more difficult when it comes to calculating the probability, but you can still use a probability tree diagram to help.
Let’s look at an example of how a tree diagram can be used to calculate probabilities when dependent events are involved.
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Greg is a baseball pitcher who throws two types of pitches, a fastball and a knuckleball. The probability of throwing a strike is different for each pitch:
Greg throws fastballs more often than he throws knuckleballs. On average, for every 10 pitches he throws, 7 of them are fastballs (probability of 0.7) and 3 of them are knuckleballs (probability of 0.3).
To find the probability that Greg will throw a strike, start by drawing a tree diagram showing the probability that he will throw a fastball or a curveball.
Probability Tree Diagram
The probability that Greg will throw a fastball is 0.7 and the probability that he will throw a curveball is 0.3. Note that the sum of the probabilities of the outcomes is 1 because 0.7 + 0.3 is 1.00.
Then, add branches for each pitch to see the probability that each pitch will be a hit, starting with the fastball:
Remember that the probability that Greg will throw a fastball for a strike is 0.6, so the probability that he will not throw it for a strike is 0.4 (because 0.6 + 0.4 = 1.00)
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Remember that the probability that Greg throws a knuckleball for a strike is 0.2, so the probability that he does not throw a strike is 0.8 (because 0.2 + 0.8 = 1.00)
Now that your probability tree diagram is complete, you can do your outcome calculations. Remember that the sum of the probability results must be equal to one:
Since you are trying to calculate the probability that Greg will throw a strike on each pitch, you should focus on the outcomes that result in a strike being thrown: a fastball for a strike or a knuckleball for a strike:
What Is A Tree Diagram
· A probability tree diagram is a useful visual tool that you can use to calculate probabilities for dependent and independent events.
· The probability of all possible outcomes must always be one. If you have a different value, go back and check for mistakes.
Check out the video lessons below to learn more about how to use tree diagrams and calculate probability in mathematics: A tree diagram can effectively illustrate conditional probabilities. Let’s start with a simple example and then look at the R code used to dynamically build a tree diagram with the data.treelibrary to show the probabilities associated with each sequential outcome.
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Gracie Skye is an ambitious girl of 10. Every Saturday she sells lemonade on the bike path behind her house during rush hour. A lot of work to prepare the stand and bring the right amount of raw material, for which she buys every Friday after school for optimal freshness.
It didn’t take long for Gracie to realize that the weather has a huge impact on potential sales. Not surprisingly, people buy more lemonade on warm, rainless days than on wet, cold days. He also estimated a demand equation based on temperature.
When it rains, the demand decreases by an additional 20% in the temperature spectrum. To create a more realistic view of his business, and to inform ingredient purchasing decisions, Gracie collected historical data to help better anticipate weather conditions.
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Gracie translates these probabilities into a tree diagram to get a better sense of all the potential outcomes and their probability. Of the
The result is no rain and a temperature of 85° F. There is a probability of 0.396 associated with this. Of the
It then uses its query function to calculate expected revenue, cost and profit for each scenario based on:
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By taking the sum of all probabilities multiplied by their associated business outcome, Gracie calculates expected values for revenue, cost, and profit for her lemonade stand operation.
I created this example because there don’t seem to be many r packages with flexible outputs for tree diagrams. In particular, I needed something with the ability to:
The objective at this stage is to create some new variables from the original inputs to help define the required tree structure.
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For us to determine the cumulative probability of a given outcome, we must multiply the probabilities in the secondary branches against the probability of the associated parent branch.
, which contains all the probabilities from our input source. We then search through the levels of the tree, taking the probabilities of all parent branches. Finally, we calculate the cumulative probability
To be able to see the final probability on the tree diagram, we need to pass data from the terminal node_type called. Because we need unique
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Values, we do this by multiplying the final branch probabilities (along with the cumulative probabilities we calculated above) and adding
And returns a tree diagram with the conditional probabilities for each path. The function can handle three additional arguments:
This approach works with tree diagrams of any size, although adding scenarios with multiple branch levels will quickly become challenging to decipher. Some things we can also consider: building a probability tree diagram is one of the ways that help us solve probability problems. In general, it is mainly used for dependent events, but we can also use it for independent events. Tree diagrams are useful for organizing and showing all possible events and their outcomes, since seeing a graphical representation of our problem often helps us see it more clearly.
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Let’s say we have $11 marbles in a bag, $4 green and $7 blue. We need to calculate the probability of:
But let’s start from the beginning. The dot on the far left represents the bag of $11 marbles. The branches represent possible outcomes. In this case, for the first drawing we will have two options:
We draw two first branches from the starting point, each representing an event. Next to them, we write the possibility of the said event.
Tree (graph Theory)
After the first draw, we inevitably had $10 marbles in the bag. It doesn’t matter what we chose in the first draw, in the second we can also choose a green or blue marble.
If we chose a green in the first draw (the first branch), for the second draw we have $3 greens left. The probability of picking a green marble is now $frac$. However, the blue marbles remain intact. The probability of choosing a blue marble is $frac$.
If we chose blue in the first lottery (second branch), we have $6 blue for the second lottery. The probability of picking blue
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