Probability Calculator

Probability Calculator 

The Probability Calculator is a versatile tool designed to simplify complex probability calculations. With its user-friendly interface, it empowers users to compute various probabilities quickly and accurately. Whether you're a student tackling statistics problems or a professional analyzing risk scenarios, this calculator provides essential functionalities. Users can input different variables, such as event outcomes, sample sizes, and probability distributions, to generate insightful results. 

From basic probability calculations to advanced statistical analyses, this tool offers valuable insights into uncertain events. With its intuitive design and robust features, the Probability Calculator is an indispensable resource for anyone working with probabilities in diverse fields like mathematics, finance, science, and engineering.

The probability calculator aids in calculating the likelihood of a single event, multiple events, two events, a series of events, and also conditional probability events. If you want to calculate the probability of A and B for any number of events, the above probability calculator will work best for you!

Well, let's get straight to the point; simply read this post to learn how to calculate probability, different probability equations, all probability formulas, statistical probability calculator, and much more that you need to know about probability.

So, let's start with the best definition of probability!

What is probability in statistics? Probability is said to be the likelihood of an event or more than one event occurring. Probability is something that indicates the possibility of acquiring a certain outcome and can be calculated using a simple probability formula.

The origin of probability theory stems from the study of games like dice, coin tossing, cards, etc. But, nowadays, probability holds great importance in decision-making. Classical theory describes that probability is the ratio of favorable cases to the total number of equally likely cases. The subjective approach reveals that the probability of an event is assigned by an individual based on the evidence available to him/her.

Definition of Probability

Probability is a way of thinking about uncertain situations and is used in a variety of fields, such as gambling, decision-making, and statistics. The definition of probability provided in this course is the most basic and fundamental understanding of the subject.

Conditional Probability

Probability is about studying randomness, and one of the most important concepts to understand is whether events are dependent or not. Two events are independent if the occurrence of the first does not affect the probability of the second. This is incredibly important because it determines how we can calculate potential outcomes. For instance, if we roll a perfectly balanced standard six-sided die, there is a probability of 1/6 of getting a two.

Although dice are linked in this example, the probability of getting a two on the second roll remains 1/6 because the events are independent. This means that the probability of getting a specific outcome, such as a two on the first roll, does not depend on what happens with the dice on the second roll.

In probability, there are different ways to view a scenario. This time, let's talk about conditional probability. Suppose you are playing a tennis match, and one of your opponents approaches the net. Depending on the angle they hit the ball, it may be possible to send the ball past your opponent in a single shot. However, if your opponent crouches when they see the ball coming, the ball is likely to bounce off the ground, and your opponent will catch it. This is an example of a situation where you think about the game in terms of events (hitting the ball) and outcomes.

Theoretical vs Experimental Probability

In most cases, theoretical probability is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes. However, there is a difference between theoretical probability and experimental probability. The formal definition of experimental probability is the ratio of the number of outcomes included in a specific category (the experiment) to the total number of outcomes. Experimental design is based on the given information, logical reasoning, and tells us what to expect from the experiment. Ideally, this information will come from the hypothesis being tested. After collecting this information, the experimental design will help you design the experiment in a way that validates or invalidates your hypothesis.

In the game of 42 marbles, a ball is randomly chosen and replaced in the bag an infinite number of times. This means that there are always 42 balls in the bag, 18 of which are orange. We can calculate the probability of choosing a particular color by dividing the number of balls of that color by the total number of balls in the bag (42). This simplifies to 3/7, or the probability is 18/42, which means that out of every 14 balls drawn, there should be 3 orange balls.

Probability is a mathematical science that deals with the likelihood of something happening. It can be used to predict what will happen as a result of conducting an experiment or to understand the chances of something happening in a given situation. In this example, we will use experimental probability to understand what happened when we drew a marble from a bag and repeated the procedure 13 more times. Suppose we have 8 orange balls in 14 attempts. This gives us the empirical probability of 8 out of 14, or 44%.

Probabilities and Statistics

Statistics is the branch of mathematics that deals with the collection, interpretation, analysis, presentation, and interpretation of data. Probability is the branch of mathematics that studies the likelihood of events and their outcomes. Understanding these differences is important as they can lead to different conclusions in different situations.

Probability is a theoretical field of mathematics that deals with things such as mathematical definitions and theorems. In contrast, statistics is a practical application of mathematics that seeks to attribute meaning and understanding to observations in the real world. Statistics can be divided into two main branches: descriptive and inferential. Descriptive statistics examines the descriptive properties of a population, such as counts, means, and standard deviations. Inferential statistics uses statistical methods to draw conclusions about a population from samples, whether from an experiment or observations taken from the real world.

Probability is the ability to predict the likelihood of events, while statistics is the study of the frequency of past occurrences. By the end of the course, you will have a deeper understanding of these concepts and be able to use them to model real-world data.

Imagine you are playing a game of chance, where each card is chosen with the same probability and your goal is to win. In this case, you could make a bet based on probabilities, namely the probability of your chosen card being a spade. Assuming the deck is complete and the choice is entirely random and fair, one could deduce that the probability is equal to 1/4. This means you can confidently place a bet.

Probability Calculator for Two Events

When you know the probability of two independent events, you can use the Probability Calculator for Two Events to determine if they occur together. You need to input the probabilities of two independent events as the probability of A and B into the calculator. Then, the calculator will display the union, intersection, and other related probabilities of two independent events along with Venn diagrams.

Probability Solver for Two Events

You can calculate the probability of various events of two independent events if you know two input values from the Probability Solver for Two Events. This is important when you don't have one or both probabilities of two events. The results will display the answer with the calculation steps.

Probability of a Series of Independent Events

You can use the Probability Calculator for a Series of Independent Events to determine the probability of each experiment containing two independent events happening one after the other. In this calculator, you need to set the number of times the event occurs.

Probability of a Normal Distribution

The normal distribution probability calculator is useful for determining the probability of a normal curve. You need to input the mean μ, the standard deviation σ, and the limits. The normal probability calculator will generate the probability of the set limits and confidence intervals for a range of confidence levels.

Introduction to Probability

Probability is the likelihood of an event occurring. When an event is certain to happen, its probability is 1. When an event is not going to happen, its probability is 0. Therefore, the probability of a given event is always between 0 and 1. The probability calculator makes probability calculations for various events incredibly simple.

Rules of Event Operations

Any grouping of the outcomes of an experiment is called an event. It is an event that can be any subset of the sample space. Complement, intersection, and union can be identified as event operation rules. Let's learn each of these rules using the following example.

Example

Your university has several faculties, including the business faculty. International students are also enrolled in this university. You have to conduct interviews with your university students as part of your project. You choose to start with the first student who walks through the door. You are aware of the following probabilities. Let's say,

A = The first student is from the Business Faculty. B = The first student is an international student.

P(A) = 0.6

P(B) = 0.3

Complementing an Event

The complement of an event is the set of all outcomes in a sample space that are not included in that event.

For example, the complement of event A means the first student is not from the business faculty. This can be denoted as A′ or Aᶜ.

Let's show the complement of event A in a Venn diagram.

Complement of Event A

In the above Venn diagram, the shaded area represents the complement of event A.

The total area of the rectangle represents the overall probability of the sample space, which is precisely equal to one. The area outside circle A shows the probability of the complement of event A. The Venn diagram allows us to establish the following relationship:

P(A) + P(A′) = 1

So,

P(A′) = 1 − P(A)

Let's find the following probabilities.

The probability that the first student you select for the interview is not from the business faculty:

P(A′) = 1 − P(A) = 1 − 0.6 = 0.4

The probability that the first student you select for the interview is not an international student:

P(B′) = 1 − P(B) = 1 − 0.3 = 0.7

Intersection of Events

The intersection of two events A and B is the list of all common elements in both events A and B. The word "AND" is often used to indicate the intersection of two sets.

The intersection of event A and event B in example 1 means selecting an international student and belonging to the business faculty. This can be denoted as A ∩ B.

Let's show the intersection of events A and B in a Venn diagram.

Intersection of Events A and B

In the above Venn diagram, the shaded area represents the intersection of events A and B.

Let's say the event of selecting a local student for the interview is C. Now, we will show events A and C in a Venn diagram.

Event A and C

The selection of an international student and also being local cannot be done simultaneously. Suppose the first student you choose is an international student. In that case, it excludes the case of the first student being a local student. Therefore, events A and C are mutually exclusive events.

Mutually exclusive events have no common elements between them. Therefore, the intersection of two mutually exclusive events is empty.

A ∩ C = φ

The probability of intersection of events can be calculated with different methods. Events A and B can be written as follows.

P(A ∩ B) = P(A) + P(B) − P(A ∪ B)

P(A ∩ B) = P(A) × P(B/A)

P(A ∩ B) = P(B) × P(A/B)

Independent Events

Independent events are events that do not influence each other. In our example, the choice of a student being from the business faculty does not affect the choice of whether they are an international student or not. Therefore, we can say that event A and event B are two independent events.

When events are independent, the probability of either of them occurring does not depend on the other. Therefore,

P(B/A) = P(B) and P(A/B) = P(A)

You can use these formulas to modify the formula we previously learned to determine the probability of the intersection of two events.

P(A ∩ B) = P(A) × P(B/A) = P(B) × P(A)

Since events A and B are independent, let's determine the probability that the first student you select for the interview is from the business faculty and is also an international student.

P(A ∩ B) = P(A) × P(B) = 0.6 × 0.3 = 0.18

Union of Events

The union of two events produces another event that contains all the elements of one or both events. The word "OR" is typically used to describe the union of two events.

In Example 1, the union of events A and B means selecting an international student or a student from the business faculty. This can be denoted as A ∪ B.

Let's show the union of events A and B in a Venn diagram.

Union of Event A and Event B

The shaded area of the above Venn diagram represents the union of events A and B.

To calculate the probability of event A or event B, we must add the probabilities of both events and subtract the probability of the intersection.

The probability of a union of events A and B can be written as follows.

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

We can modify the above formula and create a new formula to find the probability of the union of two independent events when the probability of the intersection of two events is unknown, and the two events are independent.

If the events are independent,

P(A ∩ B) = P(A) × P(B)

Therefore,

P(A ∪ B) = P(A) + P(B) − P(A) × P(B)

Let's calculate what would be the probability of combining events A and B, i.e., with what probability would we choose a student who is a business faculty student, an international student, or both at the same time?

P(A ∪ B) = P(A) + P(B) − P(A ∩ B) = 0.6 + 0.3 − 0.18 = 0.72

Thanks to the Probability Calculator for Two Events or Probability Solver for Two Events, you can easily complete all the above calculations. You can use the Probability Calculator for Two Events even if you want to verify the calculation steps as it also shows the steps for the calculation.

Normal Distribution

The normal distribution is symmetrical and bell-shaped. A normal distribution has an identical mean, median, and mode, as well as 50% of the data above the mean and 50% below the mean. The normal distribution curve moves away from the mean in both directions but never touches the x-axis. The total area under the curve is 1.

If the random variable X has a normal distribution with parameters μ and σ2, we write X ~ N(μ, σ²).

Probability of Normal Distribution

The probability density function of a normal distribution is shown below:

f(x) = 1 ​

2πσ 2

​

×e − ( x−μ) 2 2σ 2

​

In this function:

μ is the mean of the distribution; σ² is the variance of the distribution; π is 3.14; e is 2.7182.

It is impossible to provide a probability table for every combination of mean and standard deviation because there are infinitely many different normal curves. As a result, the standard normal distribution is used. The normal distribution with a mean of 0 and a standard deviation of 1 is known as the standard normal distribution.

To calculate the probability of a normal distribution, we must first transform the actual distribution into a standard normal distribution using the z-score and then use the z-table to calculate the probability. The normal probability calculator functions as a standard normal probability calculator by providing probabilities for various confidence levels.

Z = X−μ σ

The standard normal distribution curve can be used to solve a variety of real-world problems. To determine the probability of continuous variables, the normal distribution is used. A continuous variable is a variable that can take any number of values, even a decimal. Some examples of continuous variables are height, weight, and temperature.

Let's learn how to find the probability of normal distribution using the following example.

Example

The results of your group's statistics course are normally distributed, with a mean of 65 and a standard deviation of 10. Determine the probability of the following scenarios if a student is randomly selected:

the student's score is equal to or greater than 70, the student's score is less than 70, the student's score is between 50 and 70. Solution

P(X ≥ 70) = P(Z ≥ 10 70−65 ​ ) = P(Z ≥ 0.5) = 1 − 0.6915 = 0.3085

P(X < 70) = P(Z < 10 70−65 ​ ) = P(Z < 0.5) = 0.6915

P(50 > X > 70) = P( 10 50−65 ​

Z > 10 70−65 ​ ) = P(1.5 > Z > 0.5) = 0.4332 + 0.1915 = 0.6247

Calculating the probability of a normal curve involves numerous steps and requires the use of z-tables. On the other hand, the normal distribution probability calculator assists you in calculating the probability simply by entering four numbers into the calculator. To use the normal distribution calculator, you only need to input the mean, standard deviation, and left and right limits.

How to Calculate Probability (Manually, Step by Step)?

Apart from probability equations, you can simply add parameters into the previous probability calculation to find the probability of events. But if you want to calculate probability manually, read on!

All you need to follow are the steps given for probability calculation:

Firstly, you must determine a single event with a single outcome. Then, you need to identify the total number of outcomes that can occur. Next, you should divide the number of events by the number of possible outcomes. Let's delve deeper!

Step #1: Determine a Single Event with a Single Outcome: The first step in probability calculation is figuring out the probability you want to calculate. This can be indicated as an event, such as the probability of rain or rolling a specific number on a die. The event must have at least one possible outcome. For example, if you want to find the probability of rolling a three with a die on the first roll, you'll realize there's one possible outcome: either you roll a three or you don't.

Step #2: Identify the Total Number of Outcomes: Next, you need to determine the number of outcomes that can occur from the event you identified in step one. If we're talking about the example of rolling a die, there could be 6 total outcomes that can occur since there are 6 numbers on a die. So, it's clear that for an event, rolling a three, there could be 6 different outcomes that can occur.

Step #3: Divide the Number of Events by the Number of Possible Outcomes: Once you have determined the probability event along with its corresponding outcomes, you should divide the total number of events by the total number of possible outcomes. For example, rolling a die once and landing on a three can be considered the probability of one event. Therefore, each time you roll the die, it'll be counted as a single event.

How to Calculate Probability with Multiple Random Events?

If you want to calculate probability with multiple events instantly, then simply use the probability calculator for multiple events. Indeed, calculating probability with multiple random events is quite similar to calculating probability with a single event, however, there are just a few additional steps to follow to reach a final solution. The following steps highlight how to calculate the probability of multiple events:

Firstly, you need to determine each event you'll be calculating. Next, you have to calculate the probability of each event. Finally, you need to multiply all the probabilities together. 

Probability Calculator FAQ

How do you find probabilities with percentages?

If you want to calculate a probability as a percentage, you need to solve the problem as you normally would, i.e., you need to convert your answer into a percentage.

For example;

If the desired number of outcomes divided by the number of possible events is 0.25, then you need to multiply the answer by 100 to get 25%. If there's a probability of a particular outcome in percentage form, then you simply have to divide the percentage by 100 and now multiply it by the number of events to calculate the probability.

How is probability calculated on a calculator?

All you need to do is input the values in the given fields above; the probability calculator does it all for you in a couple of seconds.

What are the 3 types of probability?

The three types of probability are as follows:

Classical Relative Frequency Definition Subjective Probability What are the 5 rules of probability?

Basic Probability Rules:

Probability Rule One – (For any event A, 0 ≤ P(A) ≤ 1) Probability Rule Two: (The sum of the probabilities of all possible outcomes is said to be 1) Probability Rule Three – (The Complement Rule) Probabilities Involving Multiple Events:

Probability Rule Four – (Sum Rule for Disjoint Events) Finding P(A and B) Using Logic:

Probability Rule Five – (The General Addition Rule) How can I determine probability when choosing random numbers?

Remember it all according to the range of the random number generator. For example, if the range is from 1 to 9, then the probability of getting a specific number is said to be 1/9.

If I roll a die 6 times, what is the probability?

There's a 66.5% chance it will land on a 6 at least once.

If I rolled a standard six-sided die, what is the probability of getting a 5?

So, your answer would be 1/6, or approximately 17%.

If a six-sided die is rolled once, what is the probability of getting 1 or 2?

2/6, once the die is rolled, the probability of getting 1 is 1/6 or of getting 2 is also said to be 1/6. Therefore, 1/6 + 1/6 = 2/6 or 1/3 or 0.333.

How do I calculate probability in football matches?

Really, you can't. The only thing you can gauge is their skill. Remember players are also humans and they can have a bad day, meaning they won't play as well as usual.

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