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## Probability Problem Solver

🤖 probability solver & calculator.

This tool combines the power of mathematical computation engine that excels at solving mathematical formulas with the power of artificial intelligence large language models to parse and generate natural language answers. This creates a math problem solver that's more accurate than ChatGPT, more flexible than a math calculator, and provides answers faster than a human tutor.

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## Problem Solver Subjects

Our math problem solver that lets you input a wide variety of probability math problems and it will provide a step by step answer. This math solver excels at math word problems as well as a wide range of math subjects.

- Math Word Problems
- Pre-Algebra
- Geometry Graphing
- Trigonometry
- Precalculus
- Finite Math
- Linear Algebra

Here are example math problems within each subject that can be input into the calculator and solved. This list is constanstly growing as functionality is added to the calculator.

## Basic Math Solutions

Below are examples of basic math problems that can be solved.

- Long Arithmetic
- Rational Numbers
- Operations with Fractions
- Ratios, Proportions, Percents
- Measurement, Area, and Volume
- Factors, Fractions, and Exponents
- Unit Conversions
- Data Measurement and Statistics
- Points and Line Segments

## Math Word Problem Solutions

Math word problems require interpreting what is being asked and simplifying that into a basic math equation. Once you have the equation you can then enter that into the problem solver as a basic math or algebra question to be correctly solved. Below are math word problem examples and their simplified forms.

Word Problem: Rachel has 17 apples. She gives some to Sarah. Sarah now has 8 apples. How many apples did Rachel give her?

Simplified Equation: 17 - x = 8

Word Problem: Rhonda has 12 marbles more than Douglas. Douglas has 6 marbles more than Bertha. Rhonda has twice as many marbles as Bertha has. How many marbles does Douglas have?

Variables: Rhonda's marbles is represented by (r), Douglas' marbles is represented by (d) and Bertha's marbles is represented by (b)

Simplified Equation: {r = d + 12, d = b + 6, r = 2 × b}

Word Problem: if there are 40 cookies all together and Angela takes 10 and Brett takes 5 how many are left?

Simplified: 40 - 10 - 5

## Pre-Algebra Solutions

Below are examples of Pre-Algebra math problems that can be solved.

- Variables, Expressions, and Integers
- Simplifying and Evaluating Expressions
- Solving Equations
- Multi-Step Equations and Inequalities
- Ratios, Proportions, and Percents
- Linear Equations and Inequalities

## Algebra Solutions

Below are examples of Algebra math problems that can be solved.

- Algebra Concepts and Expressions
- Points, Lines, and Line Segments
- Simplifying Polynomials
- Factoring Polynomials
- Linear Equations
- Absolute Value Expressions and Equations
- Radical Expressions and Equations
- Systems of Equations
- Quadratic Equations
- Inequalities
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- Logarithmic Expressions and Equations
- Exponential Expressions and Equations
- Conic Sections
- Vector Spaces
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## Trigonometry Solutions

Below are examples of Trigonometry math problems that can be solved.

- Algebra Concepts and Expressions Review
- Right Triangle Trigonometry
- Radian Measure and Circular Functions
- Graphing Trigonometric Functions
- Simplifying Trigonometric Expressions
- Verifying Trigonometric Identities
- Solving Trigonometric Equations
- Complex Numbers
- Analytic Geometry in Polar Coordinates
- Exponential and Logarithmic Functions
- Vector Arithmetic

## Precalculus Solutions

Below are examples of Precalculus math problems that can be solved.

- Operations on Functions
- Rational Expressions and Equations
- Polynomial and Rational Functions
- Analytic Trigonometry
- Sequences and Series
- Analytic Geometry in Rectangular Coordinates
- Limits and an Introduction to Calculus

## Calculus Solutions

Below are examples of Calculus math problems that can be solved.

- Evaluating Limits
- Derivatives
- Applications of Differentiation
- Applications of Integration
- Techniques of Integration
- Parametric Equations and Polar Coordinates
- Differential Equations

## Statistics Solutions

Below are examples of Statistics problems that can be solved.

- Algebra Review
- Average Descriptive Statistics
- Dispersion Statistics
- Probability
- Probability Distributions
- Frequency Distribution
- Normal Distributions
- t-Distributions
- Hypothesis Testing
- Estimation and Sample Size
- Correlation and Regression

## Finite Math Solutions

Below are examples of Finite Math problems that can be solved.

- Polynomials and Expressions
- Equations and Inequalities
- Linear Functions and Points
- Systems of Linear Equations
- Mathematics of Finance
- Statistical Distributions

## Linear Algebra Solutions

Below are examples of Linear Algebra math problems that can be solved.

- Introduction to Matrices
- Linear Independence and Combinations

## Chemistry Solutions

Below are examples of Chemistry problems that can be solved.

- Unit Conversion
- Atomic Structure
- Molecules and Compounds
- Chemical Equations and Reactions
- Behavior of Gases
- Solutions and Concentrations

## Physics Solutions

Below are examples of Physics math problems that can be solved.

- Static Equilibrium
- Dynamic Equilibrium
- Kinematics Equations
- Electricity
- Thermodymanics

## Geometry Graphing Solutions

Below are examples of Geometry and graphing math problems that can be solved.

- Step By Step Graphing
- Linear Equations and Functions
- Polar Equations

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## Probability Calculator

Probability of two events.

To find out the union, intersection, and other related probabilities of two independent events.

## Probability Solver for Two Events

Please provide any 2 values below to calculate the rest probabilities of two independent events.

## Probability of a Series of Independent Events

Probability of a normal distribution.

Use the calculator below to find the area P shown in the normal distribution, as well as the confidence intervals for a range of confidence levels.

Related Standard Deviation Calculator | Sample Size Calculator | Statistics Calculator

Probability is the measure of the likelihood of an event occurring. It is quantified as a number between 0 and 1, with 1 signifying certainty, and 0 signifying that the event cannot occur. It follows that the higher the probability of an event, the more certain it is that the event will occur. In its most general case, probability can be defined numerically as the number of desired outcomes divided by the total number of outcomes. This is further affected by whether the events being studied are independent, mutually exclusive, or conditional, among other things. The calculator provided computes the probability that an event A or B does not occur, the probability A and/or B occur when they are not mutually exclusive, the probability that both event A and B occur, and the probability that either event A or event B occurs, but not both.

## Complement of A and B

Given a probability A , denoted by P(A) , it is simple to calculate the complement, or the probability that the event described by P(A) does not occur, P(A') . If, for example, P(A) = 0.65 represents the probability that Bob does not do his homework, his teacher Sally can predict the probability that Bob does his homework as follows:

P(A') = 1 - P(A) = 1 - 0.65 = 0.35

Given this scenario, there is, therefore, a 35% chance that Bob does his homework. Any P(B') would be calculated in the same manner, and it is worth noting that in the calculator above, can be independent; i.e. if P(A) = 0.65, P(B) does not necessarily have to equal 0.35 , and can equal 0.30 or some other number.

## Intersection of A and B

The intersection of events A and B , written as P(A ∩ B) or P(A AND B) is the joint probability of at least two events, shown below in a Venn diagram. In the case where A and B are mutually exclusive events, P(A ∩ B) = 0 . Consider the probability of rolling a 4 and 6 on a single roll of a die; it is not possible. These events would therefore be considered mutually exclusive. Computing P(A ∩ B) is simple if the events are independent. In this case, the probabilities of events A and B are multiplied. To find the probability that two separate rolls of a die result in 6 each time:

The calculator provided considers the case where the probabilities are independent. Calculating the probability is slightly more involved when the events are dependent, and involves an understanding of conditional probability, or the probability of event A given that event B has occurred, P(A|B) . Take the example of a bag of 10 marbles, 7 of which are black, and 3 of which are blue. Calculate the probability of drawing a black marble if a blue marble has been withdrawn without replacement (the blue marble is removed from the bag, reducing the total number of marbles in the bag):

Probability of drawing a blue marble:

P(A) = 3/10

Probability of drawing a black marble:

P(B) = 7/10

Probability of drawing a black marble given that a blue marble was drawn:

P(B|A) = 7/9

As can be seen, the probability that a black marble is drawn is affected by any previous event where a black or blue marble was drawn without replacement. Thus, if a person wanted to determine the probability of withdrawing a blue and then black marble from the bag:

Probability of drawing a blue and then black marble using the probabilities calculated above:

P(A ∩ B) = P(A) × P(B|A) = (3/10) × (7/9) = 0.2333

## Union of A and B

In probability, the union of events, P(A U B) , essentially involves the condition where any or all of the events being considered occur, shown in the Venn diagram below. Note that P(A U B) can also be written as P(A OR B) . In this case, the "inclusive OR" is being used. This means that while at least one of the conditions within the union must hold true, all conditions can be simultaneously true. There are two cases for the union of events; the events are either mutually exclusive, or the events are not mutually exclusive. In the case where the events are mutually exclusive, the calculation of the probability is simpler:

A basic example of mutually exclusive events would be the rolling of a dice, where event A is the probability that an even number is rolled, and event B is the probability that an odd number is rolled. It is clear in this case that the events are mutually exclusive since a number cannot be both even and odd, so P(A U B) would be 3/6 + 3/6 = 1 , since a standard dice only has odd and even numbers.

The calculator above computes the other case, where the events A and B are not mutually exclusive. In this case:

Using the example of rolling dice again, find the probability that an even number or a number that is a multiple of 3 is rolled. Here the set is represented by the 6 values of the dice, written as:

## Exclusive OR of A and B

Another possible scenario that the calculator above computes is P(A XOR B) , shown in the Venn diagram below. The "Exclusive OR" operation is defined as the event that A or B occurs, but not simultaneously. The equation is as follows:

As an example, imagine it is Halloween, and two buckets of candy are set outside the house, one containing Snickers, and the other containing Reese's. Multiple flashing neon signs are placed around the buckets of candy insisting that each trick-or-treater only takes one Snickers OR Reese's but not both! It is unlikely, however, that every child adheres to the flashing neon signs. Given a probability of Reese's being chosen as P(A) = 0.65 , or Snickers being chosen with P(B) = 0.349 , and a P(unlikely) = 0.001 that a child exercises restraint while considering the detriments of a potential future cavity, calculate the probability that Snickers or Reese's is chosen, but not both:

0.65 + 0.349 - 2 × 0.65 × 0.349 = 0.999 - 0.4537 = 0.5453

Therefore, there is a 54.53% chance that Snickers or Reese's is chosen, but not both.

## Normal Distribution

The normal distribution or Gaussian distribution is a continuous probability distribution that follows the function of:

where μ is the mean and σ 2 is the variance. Note that standard deviation is typically denoted as σ . Also, in the special case where μ = 0 and σ = 1 , the distribution is referred to as a standard normal distribution. Above, along with the calculator, is a diagram of a typical normal distribution curve.

The normal distribution is often used to describe and approximate any variable that tends to cluster around the mean, for example, the heights of male students in a college, the leaf sizes on a tree, the scores of a test, etc. Use the "Normal Distribution" calculator above to determine the probability of an event with a normal distribution lying between two given values (i.e. P in the diagram above); for example, the probability of the height of a male student is between 5 and 6 feet in a college. Finding P as shown in the above diagram involves standardizing the two desired values to a z-score by subtracting the given mean and dividing by the standard deviation, as well as using a Z-table to find probabilities for Z. If, for example, it is desired to find the probability that a student at a university has a height between 60 inches and 72 inches tall given a mean of 68 inches tall with a standard deviation of 4 inches, 60 and 72 inches would be standardized as such:

Given μ = 68; σ = 4 (60 - 68)/4 = -8/4 = -2 (72 - 68)/4 = 4/4 = 1

The graph above illustrates the area of interest in the normal distribution. In order to determine the probability represented by the shaded area of the graph, use the standard normal Z-table provided at the bottom of the page. Note that there are different types of standard normal Z-tables. The table below provides the probability that a statistic is between 0 and Z, where 0 is the mean in the standard normal distribution. There are also Z-tables that provide the probabilities left or right of Z, both of which can be used to calculate the desired probability by subtracting the relevant values.

For this example, to determine the probability of a value between 0 and 2, find 2 in the first column of the table, since this table by definition provides probabilities between the mean (which is 0 in the standard normal distribution) and the number of choices, in this case, 2. Note that since the value in question is 2.0, the table is read by lining up the 2 row with the 0 column, and reading the value therein. If, instead, the value in question were 2.11, the 2.1 row would be matched with the 0.01 column and the value would be 0.48257. Also, note that even though the actual value of interest is -2 on the graph, the table only provides positive values. Since the normal distribution is symmetrical, only the displacement is important, and a displacement of 0 to -2 or 0 to 2 is the same, and will have the same area under the curve. Thus, the probability of a value falling between 0 and 2 is 0.47725 , while a value between 0 and 1 has a probability of 0.34134. Since the desired area is between -2 and 1, the probabilities are added to yield 0.81859, or approximately 81.859%. Returning to the example, this means that there is an 81.859% chance in this case that a male student at the given university has a height between 60 and 72 inches.

The calculator also provides a table of confidence intervals for various confidence levels. Refer to the Sample Size Calculator for Proportions for a more detailed explanation of confidence intervals and levels. Briefly, a confidence interval is a way of estimating a population parameter that provides an interval of the parameter rather than a single value. A confidence interval is always qualified by a confidence level, usually expressed as a percentage such as 95%. It is an indicator of the reliability of the estimate.

Examples for

## Probability

Probability is the quantification of the likelihood that an event or a set of events will occur. Using Wolfram|Alpha's broad computational understanding of probability and expansive knowledge of real-world applications of probability theory, you can compute the chances of winning various games driven by random chance, conduct and analyze the experimental outcomes of random trials, visualize and compute the properties of probability distributions and calculate the probabilities of events given a set of conditions.

Compute winnings, analyze bets and determine outcomes for games of chance ranging from the toss of a coin to a game of poker and the draw of your local lottery numbers.

## Compute coin-toss probabilities:

Compute dice probabilities:, compute odds for a poker hand:, analyze a bet in roulette:, analyze a wager:, get lottery odds:.

Compute the chance people in a group have of sharing a birthday or investigate the likelihood of people's birthdays falling in a specific date range or month or on a specific day of the week.

## Compute the probability of shared birthdays in a group of people:

Specify the number of possible birthdays:, compute the probability of shared birthdays for a given interval:.

Determine the likelihood of any outcome for any number or specification of Bernoulli trials.

## Compute probabilities for a sequence of trials:

Analyze waiting-time probabilities:, find the probability of a run:.

Compute the probabilities of various compositions of events or specify individual probabilities to determine the likelihood of some, all or no events occurring.

## Compute the probability of a union of events:

Compute a conditional probability:, compute the probability of a complement:, related examples.

- Combinatorics

Compute a specific property of a probability distribution, the likelihood that an outcome has of occurring or explore the defining characteristics of a vast set of probability distributions.

## Compute properties of a probability distribution:

Specify parameters for a distribution:, compute a particular property:.

## Probability Calculator

Probability calculator finds the probability of an event occurring by calculating the ratio of the favorable outcomes to the total outcomes. Probability is a field of statistics that is used to describe the likelihood of occurrence of an event.

## What is a Probability Calculator?

Probability Calculator is an online tool that helps to determine the chance of occurrence of an event. Probability can be classified into two types, namely, experimental probability and theoretical probability. Probability is used in several industries for building predictive mathematical models. To use this probability calculator , enter values in the input boxes.

NOTE: Enter values upto 5 digits only.

## How to Use Probability Calculator?

Please follow the below steps to find the probability using the online probability calculator:

- Step 1: Go to Cuemath’s online probability calculator.
- Step 2: Enter the favorable outcomes and the total number of outcomes in the input boxes of the probability calculator.
- Step 3: Click on the "Calculate" button to find the probability of an event.
- Step 4: Click on the "Reset" button to clear the fields and enter new values.

## How Does Probability Calculator Work?

The formula for theoretical probability is given as follows:

Theoretical Probability = Number of favorable outcomes / Total number of outcomes.

There are certain axioms that are followed by both theoretical probability as well as experimental probability. These are given as follows:

- The probability of any event will always be greater than or equal to 0.
- The set of all the outcomes possible for the event is given by the sample space.
- Suppose we have two events, A and B, that cannot occur at the same time. The probability of A or B occurring is given by the summation of the probability of A and probability of B. Further, such events are known as mutually exclusive events.

The rules of probability are given below:

- The null set is used to denote the probability of an impossible event.
- The probability of occurrence of an event lies between 0 to 1.
- An event cannot have a negative probability.
- The sum of the probability of an event happening and not happening is given by 1.

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## Solved Examples on Probability Calculator

What is the probability of selecting a black ball from a bag containing 4 yellow balls, 6 blue balls, 8 red balls, and 3 black balls? Verify it using the probability calculator.

Total number of possible outcomes = 21

Favorable outcomes = 3

Probability = 3/21

Probability = 1/7 = 0.14

Thus, the probability of selecting a black ball from the given bag is 1/7 or 0.14.

If we toss a coin once, what is the probability of getting heads? Verify it using the probability calculator.

Total number of possible outcomes = 2

Favorable outcomes = 1

Probability = 1/2

Probability = 0.5

Thus, the probability of getting heads is 0.5.

Similarly, to find the probability of an event, you can try the probability calculator for the following:

- Find out the probability of getting a chocolate if a bag contains 10 chocolates and 5 toffees.
- What is the probability of getting 6 as a number when a dice is rolled?

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- High School Math Solutions – Inequalities Calculator, Exponential Inequalities Last post, we talked about how to solve logarithmic inequalities. This post, we will learn how to solve exponential... Read More

## Probability Solver

Online probability problem solver.

One of the most significant and difficult areas of statistics is probability. It focuses on the analysis of random occurrences and the probability that they will occur.

Numerous disciplines, including engineering, physics, computer science, economics, and many others, use probability. Assignments in chance can therefore be quite varied and difficult.

Based on the type of the issue, assignments in probability can be categorized into a number of different groups. There are several popular categories of probability tasks, including:

- Probability calculations: This kind of task involves estimating the likelihood that a particular event will occur. For instance, a student might be requested to determine the likelihood of rolling a particular number on a die.
- Probability Distributions: For this kind of task, probability distributions like the normal distribution, the Poisson distribution, and the binomial distribution must be used. Students might be requested to calculate a distribution’s mean, variance, and standard deviation.
- Conditional Probability: In this kind of task, you must determine the likelihood of an event given the occurrence of another event. For instance, if a student has attended all of the classes, they might be asked to estimate the likelihood that they will pass a test.
- Bayes Theorem: Working with the Bayes theorem, a mathematical formula used to estimate the likelihood of an event based on information already known, is a requirement for this kind of task. Students might be required to use the Bayes theorem to resolve a practical issue.
- Random Variables: Working with random variables and their distributions is a requirement for this kind of task. The expected value and variance of a particular random variable may be disclosed to students.
- Hypothesis Testing: In this kind of task, a hypothesis is tested using statistical techniques. For instance, a student might be requested to determine whether the population’s mean weight is equal to a specified value.

These are only a few illustrations of the various tasks in probability that students might run into. Assignments on probability can be difficult, and students may need professional assistance to finish them successfully.

Students can also check out : Best Math problem solver

Probability is not just a theoretical idea; it also plays a significant role in many practical uses. In reality, probability affects many aspects of our daily lives, often without our even being aware of it.

The following are a few examples of probability’s most typical real-world uses:

- Weather Prediction: Prediction of the weather is a well-known application of chance. To estimate the chance of precipitation, snowfall, and other weather events, meteorologists use probability. Based on past data and current circumstances, probability is used to calculate the likelihood that a specific weather event will occur.
- wagering: A key idea in wagering is probability. Probability is used by casinos and sportsbooks to establish odds for various games and sporting events. Probability is a tool that helps gamblers make educated choices about the wagers and games they should place. Probability is also used to calculate a bet’s anticipated return on investment.
- Finance: A crucial instrument in finance is probability. Probability is a tool used by stock brokers to determine the likelihood that a specific stock will appreciate or decline in value. Probability is used by insurance firms to determine the risk of insuring a specific asset or person. Probability is also used in risk management to calculate the chances of an occurrence happening and its possible effects.
- Medicine: Medical researchers use probability to evaluate the efficacy of novel therapies and medications, and it also plays a major role in medical diagnosis. Using a patient’s medical history, doctors can predict the likelihood that a patient will acquire a specific condition by using probability to diagnose diseases.
- Sports: In sports analysis, probability is a crucial instrument. Probability is a tool that sports analysts use to forecast game and competition results. The probability of a team winning a specific game or moving on to the next round of a competition is determined using probability.

It is impossible to overstate the significance of chance. From science and engineering to business and medicine, probability is used in a wide range of disciplines. Probability enables us to make well-informed choices based on the information at our disposal and to calculate the probability that a specific event will occur. Making wise choices in a variety of situations requires having a solid understanding of probability.

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In conclusion, probability tasks can be very difficult and demand a thorough knowledge of the material. Students may be given a variety of probability tasks, including those involving calculations, distributions, conditional probabilities, the Bayes theorem, random variables, and hypothesis testing. A team of highly qualified and experienced specialists who are proficient in probability work at Assignmentstore to offer the best assistance with probability homework and assignments. They provide customer assistance around-the-clock and charge reasonable rates for their services. Assignmentstore is the ideal choice if you need assistance with your chance homework.

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## Probability Calculator

This Probability Calculator computes the probability of one event, based on known probabilities of other events. And it generates an easy-to-understand report that describes the analysis step-by-step.

For help in using the calculator, read the Frequently-Asked Questions or review the Sample Problems . To understand the analysis, read the Summary Report that is produced with each computation. To learn more, read Stat Trek's tutorial on the rules of probability .

- Choose the probability you want to find in the first dropdown box.
- Identify probabilities you already know in the second dropdown box.
- Enter a known probability in each text box.
- Click the Calculate button to compute a result.
- Click the Report button to describe the analysis.

## Summary Report

This calculator uses Bayes Rule (aka, Bayes theorem, the multiplication rule of probability) to compute the probability of one event, based on known probabilities of other events.

## What is Bayes Rule?

Let A be one event; and let B be any other event from the same sample space, such that P(B) > 0. Then, Bayes rule can be expressed as:

- P(A) is the probability of Event A.
- P(B) is the probability of Event B.
- P(A|B) is the conditional probability of Event A, given Event B.
- P(B|A) is the conditional probability of Event B, given Event A.

## How to Use Bayes Rule

Bayes rule is a simple equation with just four terms. Any time that three of the four terms are known, Bayes Rule can be applied to solve for the fourth term. We've seen in the previous section how Bayes Rule can be used to solve for P(A|B). By rearranging terms, we can derive equations to solve for each of the other three terms, as shown below:

## Extensions to Bayes Rule

The terms that are required to use Bayes Rule can be computed from other probabilities. For example,

P(A) = P(A∩B) / P(B|A)

P(A) = P(A∪B) + P(A∩B) - P(B)

P(B) = P(A∩B) / P(A|B)

P(B) = P(A∪B) + P(A∩B) - P(A)

P(A|B) = P(A∩B) / P(B)

P(B|A) = P(A∩B) / P(A)

- P(A∩B) is the probability of the intersection of Events A and B.
- P(A∪B) is the probability of the union of Events A and B.

Using these formulas, Bayes Rule can be rewritten through substitution to accommodate P(A∩B) and P(A∪B) as inputs. For example, here are two "new" versions of Bayes Rule:

To compute the probability of an event, this calculator examines the known probabilities of other events and chooses an appropriate formula to complete the computation.

## Frequently-Asked Questions

Instructions: To find the answer to a frequently-asked question, simply click on the question.

## Can you explain the notation?

All of the notation used by the Probability Calculator is defined below:

- P( A ): Probability of event A
- P( B ): Probability of event B
- P( A|B ): Conditional probability of event A, given event B
- P( B|A ): Conditional probability of event B, given event A
- P(A ∪ B): Probability that event A and/or event B occurs. This is also known as the probability of the union of A and B.
- P(A ∩ B): Probability that event A and event B both occur. This is also known as the probability of the intersection of A and B.

Bayes Rule is an equation that expresses the conditional relationships between two events in the same sample space. Bayes Rule can be expressed as:

- P( A ) is the probability of Event A.
- P( B ) is the probability of Event B.
- P( A | B ) is the conditional probability of Event A, given Event B.
- P( B | A ) is the conditional probability of Event B, given Event A.

The probability calculator uses Bayes Rule to compute probabilities of one event, given probabilities of other related events.

## Can a computed probability be less than 0 or greater than 1.0?

If Event A occurs 100% of the time, the probability of its occurrence is 1.0; that is, P(A) = 1.0. And if Event A never occurs, the probability of its occurrence is 0. In the real world, an event cannot occur less than 0% of the time or occur more than 100% of the time; so a real-world event must have a probability between 0 and 1.0.

This calculator computes probabilities based on the inputs provided. It is possible to enter probabilities that could not occur together in the real world. When that happens, the calculator may generate a probability that could not occur in the real world; that is, the calculator could report a probability less than 0 or greater than 1.0.

To illustrate how this could happen, consider Bayes Rule :

- P(A) is the probability that Event A occurs.
- P(B) is the probability that Event B occurs.
- P(A|B) is the probability that A occurs, given that B occurs.
- P(B|A) is the probability that B occurs, given that A does not occur.

From this equation, we see that P(B) should never be less than P(A)*P(B|A); otherwise, the computed probability of P(A|B) will be greater than 1, which is not a valid outcome. For example, suppose you plug the following numbers into Bayes Rule:

- P(B|A) = 0.6

Given these inputs, the Probability Calculator (which uses Bayes Rule) will compute a value of 3.0 for P(A|B), clearly an invalid result. If the calculator computes a probability less than 0 or greater than 1.0, that is a warning sign. It means your probability inputs are invalid; they do not reflect real-world events.

## How can the Probability Calculator help me solve probability problems?

Solving a probability problem is a four-step process:

- Define the problem. Specify the research goal (what you want to know).
- Gather data. Collect information you need to achieve the goal.
- Analyze data. Apply the right analytical technique to achieve the research goal.
- Report results. Present the answer to the research goal.

The Probability Calculator provides a framework to help you with each critical step. From the first dropdown box, identify the probability that you wish to compute. From the second dropdown box, identify a set of probabilities that will enable you to complete the computation. Then, enter those probabilities into two or more text boxes. And finally, click the Calculate button.

The Probability Calculator does the rest. It applies the right analytical technique to the data you entered. And it creates a summary report that describes the analysis and presents the research finding.

## What is E Notation?

E notation is a way to write numbers that are too large or too small to be concisely written in a decimal format. This calculator uses E notation to express very small numbers.

With E notation, the letter E represents "times ten raised to the power of". Here is an example of a very small number written using E notation:

3.02E-12 = 3.02 * 10 -12 = 0.00000000000302

If a probability can be expressed as an ordinary decimal with fewer than 14 digits, the Probability Calculator will do so. But if a probability is very small (nearly zero) and requires a longer string of digits, the calculator will use E notation to display its value.

## Sample Problems

- Select "Find P(A ∩ B)" in the first dropdown box.
- Select "P(A), P(B), and P(A ∪ B)" in the second dropdown box
- Enter 0.25 for P(A).
- Enter 0.5 for P(B).
- Enter 0.65 for P(A ∪ B).
- Select "P(A ∩ B)" in the first dropdown box.
- From the options in the second dropdown box, we select "P(A) and P(B|A)". (We select this option, because we know these probabilities.)
- Enter 0.5 for P(A).
- Enter 0.5 for P(B|A).

Note: From the problem statement, we learned that Mary wins 75% of the time; that is, P(B) = 0.75. However, P(B) was not required to solve this problem. We only needed to know P(A) and P(B|A).

Part of the challenge in solving probability problems is distinguishing useful data from superfluous data. The Probability Calculator can help. Use the first dropdown box to choose a probability to compute. Then, use the second dropdown box to identify other probabilities that will allow you to complete the computation.

For this problem, you would select "Find P(A ∩ B)" from the first dropdown box. Then, when you look at options from the second dropdown box, you would see that one option only requires P(A) and P(B|A) to compute P(A ∩ B).

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