Probability Calculator
Calculate probabilities for single and compound events. Select the event type below and input the required values.
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Calculator Overview
This calculator is designed to evaluate three distinct levels of probability, supporting both single and compound event scenarios. A single event refers to a primary action with a solitary result, ideal for determining fundamental likelihoods. In contrast, compound events involve multiple occurrences happening simultaneously or in sequence. The calculator is particularly effective for these complex scenarios, as it automates the more intricate mathematical processes required for multi-stage outcomes.
The notation for compounded events:
- P(A): probability of Event A occurring
- P(A’): probability of Event A NOT occurring
- P(B): probability of Event B occurring
- P(B’): probability of Event B NOT occurring
- P(A∩B’): probability of only Event A occurring
- P(A’∩B): probability of only Event B occurring
- P(A∩B): probability of both occuring
- P(A∪B): probability of Event A or B or both occurring
- P(AΔB): probability of Event A or B, but not both occurring
- P((A∪B)’): probability that neither Event A or B occurring
The Math
Single Event Calculator
The single event calculator is designed to determine the likelihood of a specific result during a solitary trial. As the most fundamental type of probability tool, it focuses on the essential ratio between the number of successful outcomes and the total number of possible results.
Here is the single event probability formula without compounded successes:
P(A) = Number of Successful Outcomes / Total Number of Possible Outcomes
- Successful Outcomes: How many outcomes are favourable out of all possible outcomes
- Total Outcomes: How many possible outcomes there are
Compound success
When calculating the likelihood of consecutive successes, the individual probability is multiplied by itself for each additional trial. This process of compounding independent results determines the overall probability for the entire sequence using the following formula:
P(C) = (P(A))n
- P(A) was priorly defined and calculated
- n is the number of consecutive compound successes.
Configuration
To facilitate the calculation, you may input the total number of possible outcomes, the specific number of successful outcomes, and the frequency of consecutive successes required.
Results
The calculator provides a decimal probability reflecting the likelihood of a single event repeated across a specific frequency. Additionally, the result is expressed as a "1 in X" value to offer a more intuitive perspective on the odds. This value is determined by dividing one by the decimal probability and rounding to the nearest whole number for greater clarity.
For the probability (1 in X), the X value is calculated by dividing 1 by the decimal probability and rounding to the nearest whole number for ease of understanding.
Diagram
The visual representation for single event probability utilizes a streamlined pie chart. The green segment illustrates the proportion of successful outcomes, while the red segment identifies the proportion of non-successful outcomes relative to the entire set of possibilities.
Two Independent Events
The two independent events calculator determines the likelihood of two distinct occurrences that do not influence each other. In probability theory, events are considered independent if the outcome of the initial event has no impact on the probability of the subsequent one.
The calculation of these probabilities relies on two primary principles: the Complement Rule, which involves subtracting the probability from one, and the Product Rule, which involves multiplying individual probabilities. Because independence ensures that the occurrence of one event does not alter the probability of the other, the complements of these events remain independent as well.
P(A’): 1 − P(A)
P(B’): 1 − P(B)
P(A∩B’) = P(A) × P(B’)
P(A’∩B) = P(A’) × P(B)
P(A∩B) = P(A) × P(B)
P(A∪B) = P(A) + P(B) − P(A∩B)
P(AΔB) = P(A∪B) − P(A∩B)
P((A∪B)’) = P(A’) × P(B’)
Configuration
To facilitate the calculation, you may input the decimal probabilities for both Event A and Event B, then select your preferred result condition to generate the final analysis.
Results
Based on the selected result condition, the probability is displayed in a standard decimal format.
Diagram
The dynamic diagram utilizes a classic Venn diagram to represent the interaction between two independent events within eight potential outcomes. The specific sections of the diagram corresponding to your selected conditions will be highlighted in blue to illustrate the result.
Two Mutually Exclusive Events
The mutually exclusive events calculator identifies the probability of one of several disjoint events occurring. Events are defined as mutually exclusive if they cannot occur simultaneously; for example, a single coin toss results in either heads or tails, but never both.
Because there is no overlap between these outcomes, the mathematical process for combining them is more direct than for other event types. While your original text mentioned "dependent" events, it is important to note that mutually exclusive events are specifically characterized by their lack of common outcomes.
The calculator uses the Addition Rule for Mutually Exclusive Events. To find the probability of Event A or Event B occurring, it simply adds their individual probabilities together:
P(A∪B) = P(A) + P(B)
Configuration
To facilitate the calculation, you may input the decimal probabilities for both Event A and Event B, then select your preferred result condition to generate the final analysis.
Results
When using this calculator, you will notice specific logical outcomes that define mutually exclusive events:
- Intersection is Zero: The probability of both events happening together (P(A ∩ B)) is always 0.
- Simple Summation: In contrast to inclusive events, which require subtracting the overlap, the probability for mutually exclusive events is determined simply by summing the individual components.
- Neither event occurring: Alternative outcomes beyond Event A or B exist if the combined probabilities of P(A) and P(B) do not equal 1. Consequently, the probability of neither event occurring is determined by subtracting the union of A and B from the total probability of 1.
Diagram
The dynamic diagram for two mutually exclusive events is structured similarly to that of independent events. This calculator evaluates three specific conditions unique to mutually exclusive scenarios, with the corresponding sections of the diagram highlighted in blue to reflect the selected results.