What is a tree diagram? (And how to effectively use it)
Charts and diagrams are useful visual tools that can help better understand data. In a tree diagram, you depict an event sequence in a graphical format. While diagrams are helpful for demonstrating sales figures, charting events on a timeline or predicting trends, tree diagrams are tools that people use to calculate the probability of an outcome or outcomes. In this article, we discuss the definition of a probability tree diagram, explain how to calculate probabilities with it and provide tips for making your calculations successfully.
What is a probability tree diagram?
Probability tree diagrams are often used to calculate the number of possible outcomes of an event and the probability that they could occur. The design of this diagram also visually organises these results into a tree configuration. This tree has 'branches' and each branch represents a different probable outcome. You can figure out the likelihood of a certain series of events by multiplying probabilities along the branches. To ensure that your work is accurate, ensure that all final probabilities add up to 1.0.
If you use a coin toss as a simple example, the first two branches represent the outcomes of your first coin flip. There are only two possible outcomes, heads or tails, and the probability of either is 0.5. You can fill in further tree branches and probabilities to represent the possible outcome every time you might flip the coin again.
Why are probability tree diagrams important?
The requirement to calculate the probable outcome of an event is of crucial importance to many professions. Here are some professionals that benefit from utilising probability trees:
Meteorologists: the probability tree can be useful for those in this field to analyse weather patterns and predict the probability of specific weather conditions.
Epidemiologists: probability is an essential part of detecting the source and spread of disease, and these professionals rely on it to calculate and analyse the connection between disease exposure and contraction.
Statisticians: often responsible for managing financial risk and probabilities, these professionals collect and analyse data to assist businesses in making high-level decisions.
Cost estimators: probability tree diagrams are helpful for those in this role to consider how to reduce costs, time and resources spent in making or providing a product.
Insurance underwriters: the role of these professionals is to evaluate the risk of an insurance client. The process becomes easier when they use this method to calculate the probability of risk.
Market research analysts: these analysts research and gather data on consumer markets to help understand trends in products and services, implementing probability helps them to predict consumer behaviour.
Astronomers: these professionals collect data from distant space to help understand the cosmos. They use probability to help predict occurrences like solar flares, meteor showers and comet orbits.
Sportscasters: those working in the arena of sports can study patterns of probability to analyse and assess the outcomes of events and forecast future sports outcomes for their audiences.
Physicians: these practitioners predict medical outcomes for their patients. They rely on probability to assess health issues such as pregnancy, life expectancy or the duration of an illness.
Math teachers: to help prepare their students for making everyday decisions and addressing life situations, teachers train them to use probability to help decide the most suitable course of action to take.
How to calculate probabilities with tree diagrams
While there are many techniques used to calculate probabilities, an advantage of using this diagram type is to help organise your data into useful visual aids. Follow these six steps to help calculate the probability of an event occurring:
1. Outline the possible outcomes
First, establish the answer you're looking for and outline the possible outcomes. For example, you're trying to assess if your soccer team, the Reds, can win their next game. This outcome may depend on who your opponents are, the Blues, who are highly skilled, or the Yellows, a team with a lot of novice players.
You first create a dot, then draw two arrows pointing away from it. As you could play either of these teams, write the 'Blues' and the 'Yellows' at the ends of these lines. Next, you can write the probability of each outcome on the arrow's line.
2. Write the probability of each outcome
On average, you play the Yellows about six times every 10 games, so the probability that you might play the Yellows today is 0.6. You next subtract 0.6 from 1 to find the probability of playing the Blues, which is 0.4. Now write 0.6 on the arrow that points to the Blues and on the arrow that points to the Yellows, write 0.4. Complete a quick tally of 0.6 and 0.4 to make sure they equal 1.0.
3. Create your next tree branches
Last season against the Yellows, you won eight out of 10 games, which means if you play the Yellows today, there's a 0.8 probability chance of you winning. As you lost to the Yellows two out of 10 games last season, it also means the probability of the Reds losing is 0.2. On your tree, create two new arrows branching to the right from the 'Yellows' branch head. Each one leads to 'win' and 'lose' outcomes. On the 'win' arrow, write 0.8. On the 'lose' arrow, write 0.2.
The games against the Blues last season resulted in your team winning five out of 10 games. This means that you also lost five out of 10 times. On your diagram now, create two new arrows facing the right, leading to 'win' and 'lose'; outcomes after the 'Blues' branch head. On the 'win' arrow, write 0.5 and on the 'lose' arrow, you can also write 0.5. Note the sum of 0.5 and 0.5 to still adds up to 1.0.
4. Calculate the overall probabilities
Your objective is to figure out if your team might win the game. You now have the data to calculate the overall probabilities by multiplying each branch of the tree along the way. First, multiply the probability that you play the Yellows, 0.6, by the probability that you win against the Yellows, 0.8. The result gives you 0.48 of winning against the Yellows in today's game. Next, multiply the probability that you play the Blues, 0.4, by the probability that you win against the Blues, 0.5. This gives you a 0.20 chance of winning against the Blues in today's game.
5. Add the relevant probabilities
To predict the general likelihood that you can win today's game, you can add the probabilities in the 'column' of the tree to calculate your final result. By adding 0.48, the chance you win against the Yellows, with 0.20, the chance you win against the Blues, gives you a result of 0.68. This indicates the overall probability that you might win the game. To see how this translates to a percentage chance of winning, multiply 0.68 by 100. This concludes that the Reds have a 68% chance of winning today's soccer game.
6. Check that your work is right
To be thorough, check your work by calculating the overall probabilities of losing the game and then add up all the numbers in the column to ensure they equal 1.0. To calculate the probability of losing to the Yellows in today's game, multiply 0.6 by 0.2, which equals 0.12. Repeat this formula for losing to the Blues in today's game by multiplying 0.4 by 0.5, which equals 0.20.
Next, tally both probabilities 0.12 and 0.20 to get 0.32. This is the overall probability of losing the game today. Finally, add 0.32 with 0.68, which determines the probability of winning and you get 1.0. This checking method reassures you have performed your calculations correctly.
Tips for calculating the probability
You may find this tool helpful in your daily work, whether you're a student or professional. Here are some tips for effectively calculating probability using a tree diagram:
Begin with paper: at an early stage, it's a good idea to create a simple diagram using a pen and paper. Writing down information and creating a chart helps you to understand its function.
Collect all preliminary data: prior to drawing a diagram, it's a good idea to list every piece of information you intend to use. This includes the outcomes you're measuring and the probabilities you already know.
Pay close attention to the numbers: attention to detail is essential when calculating statistics. Be methodical when applying the formulas to the numbers you use and be certain each step along the way has the right value.
Re-check your work: it's a good idea to check the accuracy of your work using a checking method when performing any mathematical calculation.
Probability equals estimate: a tree diagram is a helpful tool for predicting outcomes, but it doesn't guarantee its results to become outcomes. Unexpected outcomes are inevitable, so try to manage your expectations appropriately.
Ask for help: it may take you a while to master the use of this diagram, so find a teacher, colleague or mentor who can assist and guide you through the process.