What is the Probability Tree? 

Probability theory is a branch of mathematics that deals with the subject of probability. Although there are many different concepts of probability, probability theory expresses the definition mathematically through a series of axioms. Usually, these axioms express probability in terms of a probability space, which assigns a measure with values ranging from 0 to 1 to a set of outcomes known as the sample space. An event is a subset of these outcomes that is described. 

A tree diagram can be used to describe a probability space in probability theory. For calculating combined probabilities, probability trees are useful. It allows you to visually map out the odds of several different situations without having to use complex probability equations. At the end of each branch in the tree diagram, you'll find the result of a specific case. 

Use of a Probability Tree

It's not always clear whether to multiply or add probabilities. It's easier to find out when to add and when to multiply with the help of a probability tree. Plus, rather than a lot of equations and numbers on a sheet of paper, seeing a graph of your problem will help you see the problem more clearly. 

Parts of a Probability Tree 

• The branches and the ends are the two key components of a probability tree. Each branch's probability is usually written on the branches, while the result is written on the branches' ends. 
• A series of independent events (such as a series of coin flips) or conditional probabilities may be represented using tree diagrams (such as drawing cards from a deck, without replacing the cards). Each node on the diagram represents an occurrence and has a probability associated with it. Since the root node represents a specific event, it has a higher probability. 
• Tree diagrams begin with a parent or head event and branch out into additional potential events, each with a percentage or probability. 
• The cumulative likelihood of the sequence of events actually happening is calculated by multiplying the branches; all probabilities added together should equal 1.0. 

Conditional Probability 

The probability of an event or outcome occurring based on the occurrence of a prior event or outcome is known as conditional probability. The modified probability of the following, or conditional, event is multiplied by the probability of the preceding, or conditional, event to get conditional probability. 

Consider the following scenario: a fair die has been rolled, and you are asked to estimate the likelihood that the result was a five. Your response is 1/6 because there are six equally possible outcomes. Consider what would happen if you were given additional details before answering, such as the fact that the number rolled was odd. Since there are only three odd numbers that can occur, one of which is five, you'd have to update the probability calculation that a five was rolled from 1/6 to 1/3. 

Conditional Probability Formula 

P(B|A) = P(A and B) / P(A) 

Or 

P(B|A) = P(A∩B) / P(A) 

Bayes' Theorem 

The Bayes theorem is a mathematical formula for calculating conditional probability, named after 18th-century British mathematician Thomas Bayes. The theorem allows you to revise existing predictions or hypotheses (update probabilities). This set of probability rules helps one to update their forecasts of future events based on new knowledge, resulting in more precise and dynamic estimates. 

Independent Events 

Independent events have no impact on the occurrence or probability of other events, and their likelihood is unaffected by or influenced by the occurrence of other events. 

Independent events are those whose occurrence is not based on the occurrence of another event. A and B are said to be independent events if the likelihood of occurrence of one event A is not influenced by the occurrence of another event B. 

Starting a Tree Diagram 

Each tree diagram begins with the parent, which is the first event. The outcomes are extracted from the parent case. Let's use the example of flipping a coin to keep it as straightforward as possible. The parent case is the process of flipping the coin. 

There are two potential outcomes from there: drawing heads or tails. This is how the tree diagram will look: 

To account for any additional probabilities, the tree can be expanded almost indefinitely. Consider the following scenario: 

A second coin flip is represented by the second string of possibilities; the first may be either heads or tails. However, if the first toss shows heads, there are two potential outcomes for the second toss, and the same if the first toss shows tails. Now it's time to work out the odds. 

Calculating Probabilities with a Tree Diagram 

The line drawn from one arrow to the next represents each branch on the tree. Since there are only two possible outcomes when tossing a coin, each outcome has a 50 percent (or 0.5) probability of occurring. As a consequence, the chance of flipping tail, then tail again in the example above is 0.25 (0.5 x 0.5 = 0.25). The same can be said for: 

• Tail, then head 
• Head, then tail 
• Head, then head 

Add the list of total probabilities to ensure that the probabilities are right.  

0.25 + 0.25 + 0.25 + 0.25 = 1.0 in this case. When all odds are added together, they should equal 1.0. 

Binomial Probability 

A binomial distribution can be thought of as the likelihood of a success or failure outcome in a multiple-repeated experiment or sample. The binomial distribution is a type of probability distribution with two possible outcomes (the prefix "bi" means "two" or "twice"). 

The probability of exactly x successes on n repeated trials in an experiment with two possible outcomes is known as a binomial probability (commonly called a binomial experiment). The binomial probability is if the probability of success on an individual trial is p, then the formula is given as: 

_nC_x⋅p^x⋅(1−p)^n−x   

Where 

• _nC_x indicates the different combinations of x objects selected from a set of n objects x. 
• p is the probability of a single trial's success, and (1-p) is the probability of a single trial's failure. 

Probability Using a Venn Diagram

A Venn diagram is often used to visualize the probabilities of different events. The use of a Venn diagram to determine the probabilities of individual events, the intersection of events, and the complement of an occurrence is discussed in the following diagram. 

Let's look at the Venn diagram below to see what the possibilities are. 

It's important to note that the sum of all the values in the diagram is: 0.4+0.3+0.2+0.1=1 

The entire sample space for two cases, A and B, is shown in this diagram. We can find the probabilities of the events that occur as shown. 

P(A) : To find the P(A), we will add the probability that only A occurs to the probability that A and B occur to get 0.4+0.3=0.7. So P(A)=0.7. 

P(B) : Similarly, P(B)=0.2+0.3=0.5. 

P(A∩B) : Now, P(A∩B) is the value in the overlapping region 0.3. 

P(A∪B) : P(A∪B)=0.4+0.3+0.2=0.9.  

Common Mistakes 

• Some students may give some probabilities greater than one, which suggests that they did not fully comprehend the laws and concepts of probability. 
• Students might add the probabilities of the consecutive branches instead of multiplying them. 

Context & Applications 

• They are used by scientists and statisticians in a number of fields, as well as by government agencies. 
• Financial mathematics, insurance, industrial quality control, biology, quantum mechanics, and the kinetic theory of gases all benefit from this expertise of probability theory. 

Related Concepts 

• Binomial distribution 
• Mutually exclusive events