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# Probability Tree

It’s not always clear whether to multiply or add probability. It’s easier to figure out when to add and when to multiply with the help of a probability tree. Furthermore, rather than a lot of formulas and data on a piece of paper, visualizing a graph of your problem will help us see the situation more simply. A tree chart is a diagram used in mathematics, specifically in probability and statistics, to aid in the calculation and visualization of probabilities. Just at end of the branches in the tree diagram, we’ll find the probability of a specific event.

## Definition:-

Tree diagrams are a visual representation of the interactions between two or more events. At the conclusion of each branch, the outcome is labeled, and the probability is written alongside the line. Here the question arise for probability so, first define probability

Probability is a metric for determining the possibility of an event occurring. Many things are impossible to forecast with 100% accuracy. Using it, we can only anticipate the probability of an event occurring, i.e. how probable it is to occur. Probability must ranges from zero to 1, with 0 indicating an improbable event and 1 indicating a certain event. Probability for Class 10 is an important subject for students because it teaches all of the fundamental ideas. All of the occurrences in a sample space have the same probability.

If we flip a coin, as illustration, we can obtain either Head OR Tail; there are only two possible outcomes (H, T). If the chance of the very first occurrence has no bearing on the chance of the second event occurring, the two events are considered independent.

The probability of getting a 6 on a die, as particular, has no bearing on the probability of rolling a 6 the next time. Each roll’s score is distinct from the others.

If the chance of the very first occurrence has no bearing on the chance of the second event occurring, the two events are considered independent.

### Example

The probability of getting a 6 on a die, as particular, has no bearing on the probability of rolling a 6 the next time. Each roll’s score is distinct from the others.

If the chance of the very first occurrence has no bearing on the chance of the second event occurring, the two events are considered independent.

The probability of getting a 6 on a die, as particular, has no bearing on the probability of rolling a 6 the next time. Every roll’s score is distinct from the others.

The tree diagram aids in the organization and visualization of various alternative possibilities. The two primary positions of the tree are the branches and the ends. Every division’s probability is displayed on the branch, while the ends provide the final conclusion. When determining when to multiply and when to join, tree diagrams are employed.

## A Probability Tree Diagram’s Components

The branches and the endpoints are the two fundamental components of a probability tree (sometimes called leaves). Each branch’s probability is normally placed on the branching, whereas the outcome is written on the branches’ ends.

## Different Types of Occurrences

In most tree diagrams, there seem to be two sorts of occurrences shown. They are as follows:

### 1. Probabilities with conditions

Conditional probabilities, often known as “dependent events,” are the enhanced possibilities of an event occurring because another event has already occurred. Conditional (depending) events, but at the other hand, usually only happen if/when other events happen.

### 2. Independent occurrences

Definition:-dependence occurrences have no bearing on the occurrence or probability of other events, and their likelihood of occurring is unaffected by the occurrence of many other occurrences.

It is frequently easier to solve an issue without utilizing tree diagrams if you already know that events are independent. However, a tree diagram should be used if you are unsure whether events are independent or if you know they are not.

### Examples

Use the following as examples of separate events:

Having a dog and cultivating your own herb garden are two of life’s greatest pleasures.

Having a Honda Corvette and paying off your mortgage early.

Winning the lotto and running out of milk are two scenarios that come to mind.

Purchasing a lottery ticket and then discovering a penny on the ground (your odds of finding a penny doesn’t depend on a buying lottery ticket).

Getting home in a cab and watch your favorite movie on television.

Receiving a parking fine and going to the casino to play dice

## Steps to make tree diagram

In such a probability tree diagram, the rule for determining the chance of a specific event occurring is to combine the probabilities of the relevant branches.

Phase 1: Line is drawn to indicate the question’s initial set of alternatives  Label them as follows: We’ll use the letters A, B, and C from our query.

Phase 2: Transform the percentages to decimals and arrange them on the diagram’s corresponding branch. In our case, 50% equals 0.5, and 25% equals 0.25.

Phase 3: Add the next set of branches to the tree. In our case, we were told that passenger vehicles accounted for 70% of firm A’s total. Thus have 0.7 P (“P” is just my own abbreviation here for “Passenger”) and 0.3 NP (“NP” = “Not Passenger”) when we translate to decimals.

Phase 4: Repeat step 3 as needed for as many branches as you have.

Phase 5: Add the probability of the first and second branches that generate the desired result. In our example, we’re interested in passenger plane manufacture, so we chose the first branches that lead to P.

Phase 6: Combine the remaining branches to get the result we want. There are two more branches that can lead to P in our case.

Phase 7: Add all of the probabilities from steps 5 and 6 together.

Every tree diagram begins with the parent, which is the first occurrence. The outcomes are derived from the parent event. Let’s use the example of flipping a coin to keep things as simple as possible. The parent event is the act of flipping the coin.

There really are two conceivable outcomes from there: pulling heads or tails. This is how the tree diagram can focus:

## Using a Tree Diagram to Calculate Probabilities

Probabilities are usually calculated using addition or multiplication. Knowing what to do and when is, nonetheless, critical. Let’s look at the previous case.

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

T, H

H, THEN T

H, H

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

### Example

Let’s derive tree diagram of tossing a coin two times

Hence, event can be written as {HH, HT,TH, TT}

## Calculation of Probabilities

Likelihood are usually calculated using addition or multiplication. Knowing where to go and what is, nonetheless, critical. Let’s look at the previous case.

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

Include the table of total probabilities to ensure that the probabilities are accurate. 0.25 + 0.25 + 0.25 + 0.25 = 1.0, in this situation, once all probability is summed together, they must equal 1.0.

## To determine conditional probabilities, use tree diagrams.

While dealing with conditional probabilities, it’s beneficial to visualize the likelihood of various outcomes using a tree diagram. Let’s review the conditional probability formula to better grasp how tree diagrams are employed.

### Definition:

Probability with Conditions

Assuming that event A already has happened, the chance of event B occurring is

P(B/A) = P(A∩B)/P(A)

When P(B/A) is the likelihood of B because of this A occurs, (P(A∩B) is the likelihood of A and B  taking place at the same moment, and P(A)  is the likelihood of A  Occurring.

### Example

The version of the formula can be used to calculate the likelihood of two occurrences colliding. It can be estimated using a tree diagram, with the first tree indicating the likelihood of A and the second branch representing the probability of B provided that A has occurred, by multiplying across branches.

We can implement such probability principles to a tree diagram that use these rules. Because the odds of all conceivable events sum up to 1, each set of branches in a tree diagram must add up to 1. As just a result, if we have two outcomes of an event, with the probability of A as the first branch of the pair, the probability of A, the complement of A, must be the second branch of a series. Similarly, if we have the probability of B provided that A is one of the pair of trees, then the likelihood of B there’s the other branch, therefore the likelihood of B’ given that A has occurred must be the other branch. Furthermore, if B is a possibility, given that A’ has occurred as one of the two branches, the likelihood of B’ must be the opposite branch given that A’ has happened. The next tree diagram can also be used to demonstrate it.

WE can use the counterpart to find the other branch because the tree diagram previous section only has pairs of branches. That’s not the case, however, if there are three or more sets of branches.

A branch diagram is a graphical technique of illustrating a probability problem when there are a relatively small number of outcomes for compound (more than one) events.

We put the probability of each result on each branch of the tree, and the preceding should hold true in every tree diagram:

For each pair of branches, the sum of the probability should equal 1.

The total of all end options’ probability should also equal 1.

## QUESTION

If the rains on one day, there is a 1/4 chance that it will rain the next day. When it does not rain on a given day, there is a 1/2 chance that it will rain the next day. It’s a 1/7 chance that it’ll rain tomorrow. What are the chances of it raining the day after tomorrow? To figure out the solution, make a tree diagram of all the choices.

STEP 1:- Construct the tree diagram’s initial level.

Until we can figure out what will happen the day after tomorrow, we must first figure out what will happen tomorrow. We’ve been advised that there’s a 15 percent chance of rain tomorrow. And here is the how to use a tree diagram to describe this data

Construct the tree diagram’s second level.

We’ve also been warned that if it rains one day, it will almost certainly rain the next. On either hand, if it does not rain on one day, there is just a chance that it will rain the next day. We create the tree diagram with this information:-

## Calculate the probability.

We’re asked whatever the chances are of it raining the day after tomorrow. We can see from the tree diagram above that there are two scenarios in which it rains just after next.

Probability f rain =  (1/7) x (1/4)

= 1/28

Probability of no rain = (6/7) x (1/2)

= 6/14

Hence probability of rain tomorrow = (1/28) + (6/14)

= 13/26

= ½ probability of raining. ,

Hence, A probability tree would be exceedingly untidy in more sophisticated cases, but by recalling how a probability tree is built, we may compute the likelihood of occurrence without physically creating a tree.

A tree structure is a sort of graph that is used to calculate the results of the experiment. It is made up of “branch offices” that are labeled with probability or frequencies. Some probability problems can be visualized and solved more easily using tree diagrams.

## How does the tree diagram function properly?

The idea behind a tree diagram is to start with the whole thing, or one, on the left side. When there are multiple possible possibilities, the likelihood in that branch breaks into smaller branches for each.

The graphic begins with a single node and branches out to other nodes that indicate mutually exclusive decisions or events. The investigation will start at first blank line in the picture below. Node A or B will thereafter be reached as a result of a decision or occurrence. Additional decisions or events will occur as a result of these secondary nodes, continuing to the third level of endpoints until a consensus is drawn.

In additional to math, tree diagrams are utilized in decision making, business valuations, or chance calculations. Tree diagrams include a strategic answer by combining the probabilities, decisions, costs, and payouts of a decision. Specific the market value of the underlying securities at a given point in time, we can use a decision tree to represent the price of a put or call option in finance. Decision trees are increasingly being employed in the design of fetch algorithms and the customer experience of financial apps. One application of decision trees is determining an appropriate investment plan for a new rob advisor user based on an orientation form.

## Summary

• In math’s, tree diagrams are used to show the likelihood of certain occurrences occurring; events are either dependent – one cannot happen without another – or independence – one does not impact the other.
• Tree diagrams begin with a parent or head event and branch out into numerous possible occurrences, each with a proportion of chance.
• The overall probability of the sequence of events actually happening is calculated by multiplying the branch offices; all likelihoods added together will equal 1.0.
• A decision tree is simple to comprehend and use.
• Expert advice and opinions, as well as hard evidence, can be incorporated.
• It can be combined with other decision-making methods.
• It’s simple to add branching storylines.
• When attribute values with many levels are utilized in a decision tree, the factors with more levels receive more information.
• Simulations can quickly grow complicated; however this is typically only an issue if indeed the tree is being built by manually.
• When it comes to interpreting dependent events, decision trees are extremely useful. You can use a tree diagram to explain how another possible outcome of one event influences the odds of the other occurrences.
• With independent occurrences, tree diagrams are less informative because we can simply multiply the probabilities of distinct events to find the likelihood of the merged event.