Law Of Total Probability Calculator

Enter probabilities of events A₁, A₂, …, An (comma-separated):


Enter conditional probabilities P(B|A₁), P(B|A₂), …, P(B|An) (comma-separated):


Calculate

In probability theory, it’s often useful—or necessary—to calculate the probability of an event based on conditional probabilities and a partition of the sample space. This is where the Law of Total Probability becomes a powerful tool.

The Law of Total Probability Calculator simplifies this process by automating the calculation, even for multiple partitions. Whether you’re a student preparing for an exam, a data scientist working on a probabilistic model, or just someone interested in conditional events, this tool will help you get accurate results quickly.

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Formula

The Law of Total Probability is used when an event B depends on the occurrence of one of several mutually exclusive and exhaustive events A₁, A₂, …, An. The formula is:

P(B) = P(B|A₁)·P(A₁) + P(B|A₂)·P(A₂) + … + P(B|An)·P(An)

This equation gives the total probability of B occurring, based on how likely B is under each partition Aᵢ and the probabilities of those Aᵢ themselves.

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How to Use the Calculator

This tool allows you to input:

1. A list of P(Aᵢ) values (probabilities of each event A₁, A₂, …, An).
2. A list of corresponding P(B|Aᵢ) values (conditional probabilities of B given Aᵢ).

Steps:

• Input each list as comma-separated values.
  Example:
  • P(Aᵢ): 0.2, 0.3, 0.5
  • P(B|Aᵢ): 0.6, 0.7, 0.8
• Click “Calculate”
• Get your result instantly as P(B)

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Example

Given:

• P(A₁) = 0.3, P(A₂) = 0.7
• P(B|A₁) = 0.5, P(B|A₂) = 0.9

Solution:

P(B) = (0.5 × 0.3) + (0.9 × 0.7)
P(B) = 0.15 + 0.63 = 0.78

✅ Final Probability: 78%

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Applications

The Law of Total Probability is widely used in fields such as:

• Machine Learning: Modeling conditional dependencies.
• Statistics: Partitioning complex events.
• Bayesian Inference: Setting up prior distributions.
• Decision Trees: Calculating leaf probabilities.
• Medical Diagnostics: Estimating test probabilities under conditions.

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FAQs

1. What is the Law of Total Probability?
It states that if events A₁, A₂, …, An form a partition of the sample space, then the total probability of B is the weighted sum of P(B|Aᵢ)·P(Aᵢ).

2. Can the probabilities exceed 1?
No. All individual probabilities (P(Aᵢ), P(B|Aᵢ)) must be between 0 and 1.

3. Do P(Aᵢ) values need to add up to 1?
Ideally yes, since A₁ through An form a partition of the sample space.

4. How many partitions can I enter?
As many as you like—just ensure the two input lists are the same length.

5. Can this be used for conditional events?
Yes, this calculator is built for exactly that: evaluating the total probability based on conditionals.

6. What if the lists are different lengths?
The calculator will show an error—you must enter an equal number of values.

7. Can I use percentages?
No. Convert percentages to decimals first (e.g., 40% → 0.4).

8. Can I get step-by-step working?
This version gives you the total result. For steps, we can make a detailed output version.

9. What if I make a mistake in formatting?
You’ll get a helpful error message prompting correction.

10. Can this help with Bayesian Theorem problems?
Yes—this is often used as a preliminary step in applying Bayes’ Theorem.

11. What’s the real-world use case?
Anywhere probabilities are broken into cases—diagnostics, marketing, ML models, etc.

12. Can I include negative numbers?
No—probabilities cannot be negative.

13. Does the calculator round results?
Yes, to 6 decimal places.

14. Is this mobile friendly?
Yes, it works on smartphones and tablets.

15. What if I want the probability of B not happening?
Just subtract: P(B’) = 1 – P(B).

16. Can I use this for decision analysis?
Yes! It’s very helpful for computing weighted likelihoods across branches.

17. Is this calculator free?
Yes, you can use it anytime.

18. Can I embed this on my site?
Yes! Contact us for a copy/paste HTML version.

19. Does it support multiple languages?
Currently English only, but can be translated.

20. How is this different from Bayes’ Theorem?
This computes P(B) directly. Bayes’ Theorem uses P(B) in its denominator to compute P(Aᵢ|B).

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Conclusion

The Law of Total Probability Calculator provides a fast, reliable way to compute complex event probabilities by partitioning the sample space. It’s ideal for academics, analysts, and professionals working with uncertainty. By automating this essential concept, the tool helps make informed decisions and simplifies multi-scenario probability modeling.