Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Probability and Statistics → Probability Theory ↓


Conditional Probability and Bayes' Theorem


Probability theory is an important part of mathematics that studies the likelihood of outcomes given certain known conditions. Among the many concepts within this field, conditional probability and Bayes' theorem hold an important place, allowing us to adjust our initial beliefs based on new data or observations.

Understanding conditional probability

Conditional probability refers to the likelihood of an event occurring, provided that another event has already occurred. To understand this concept, it is necessary to distinguish it from regular probability which measures the likelihood of an event occurring out of all possible outcomes. On the other hand, conditional probability is the probability relative to another event.

Formula of conditional probability

Let us denote two events as (A) and (B). The conditional probability of (A), given that (B) has occurred, is denoted as (P(A mid B)). The formula to calculate it is:

P(A mid B) = frac{P(A cap B)}{P(B)}

Here:

  • (P(A cap B)) is the probability that both events (A) and (B) occur.
  • (P(B)) is the probability that the event (B) occurs.

This formula is valid assuming that the probability of (B) ((P(B))) is not zero. It essentially tells us how to adjust our probabilities based on the occurrence of another event.

Example of conditional probability

Imagine a deck of cards with a total of 52 cards. This deck has 12 face cards (kings, queens, jacks) and 4 aces. What is the probability that a queen is drawn if the card drawn is a face card?

To solve this question:

  • The probability of drawing a face card, (P(B)), is:
    • P(B) = frac{12}{52}
  • The probability of drawing a queen from all face cards, (P(A cap B)), assuming there are 4 queens, is:
    • P(A cap B) = frac{4}{52}
  • Thus, the conditional probability P(A mid B) is:
    • P(A mid B) = frac{P(A cap B)}{P(B)} = frac{frac{4}{52}}{frac{12}{52}} = frac{1}{3}

This example shows how to evaluate the conditional probability (frac{1}{3}), which means that one third of the face cards are queens.

Discovery of Bayes' theorem

Bayes' theorem plays an important role in statistics and decision making. It connects the conditional probability of two events (A) and (B), essentially reversing the conditioning provided by earlier data.

Formula of Bayes theorem

The equation of Bayes theorem is presented as follows:

P(A mid B) = frac{P(B mid A) cdot P(A)}{P(B)}

Where:

  • (P(A mid B)) is the probability of (A) given (B). This is what we are trying to find.
  • (P(B mid A)) is the probability of (B) given (A).
  • (P(A)) is the probability of (A) occurring.
  • (P(B)) is the probability of (B) occurring.

Application of Bayes' theorem: Text example

For example, suppose there is a medical test for a disease that is 99% accurate. This means that if someone has the disease, the test will be positive 99% of the time. However, there is a 1% false-positive rate, where the test is positive even when the person does not have the disease. Suppose that 0.5% of the population actually has the disease. If a test subject tests positive, what is the probability that they actually have the disease?

We define:

  • Event (A): The subject has a disease.
  • Event (B): The test result is positive.

From the problem we get:

  • The probability of the test being positive, given the subject has the disease ((P(B mid A))), is 0.99 (99% accuracy).
  • The probability of the test being positive, provided the subject does not have the disease, is the false positive rate: 0.01.
  • The probability that the subject has the disease ((P(A))) is 0.005 (0.5%).

We need to find (P(A mid B)), for which we need to calculate (P(B)). Using the law of total probability, we calculate:

P(B) = P(B mid A) cdot P(A) + P(B mid neg A) cdot P(neg A)

where (P(neg A)) is the probability that the subject does not have the disease:

P(neg A) = 1 - P(A) = 0.995

like this:

P(B) = 0.99 times 0.005 + 0.01 times 0.995 = 0.01485

Now apply Bayes' theorem:

P(A mid B) = frac{P(B mid A) cdot P(A)}{P(B)} = frac{0.99 times 0.005}{0.01485} approx 0.333

Therefore, even after a positive test result, there is still a chance of about 33.3% that the subject actually has the disease. This highlights the importance of interpreting test results in the context of the prevalence of the condition and the accuracy of the test.

Visual example

To understand Bayesian updating visually, consider the following illustration of how information is updated. Suppose we are in a scenario with a set of possible outcomes, represented by circles. Our current beliefs before we see any evidence are shown as prior probabilities. The introduction of new evidence, represented by overlapping or aligned layers, gives us updated or posterior probabilities. This interaction highlights how evidence changes our beliefs in the Bayesian framework.


    
        
        East
        
        
        Proof
        
        
        Back

In this illustration, the intersection of the prior and evidence circles gives rise to the posterior distribution. This is a simplified way of looking at how prior knowledge and new evidence combine using Bayes' theorem.

Further findings and implications

Understanding conditional probability and Bayes' theorem lays the groundwork for many advanced topics in statistics and machine learning, such as Bayesian networks and inferential statistics. In these fields, we constantly update our beliefs as more data or evidence becomes available, something that Bayes' theorem allows us to do widely.

Moreover, both concepts are important in decision making, allowing us to make informed decisions based on current information rather than static assumptions. They represent a dynamic method for dealing with problems, which contributes to processes that require constant refinement of probability assessments.

Thus, whether dealing with risk assessment, scientific forecasting, or artificial intelligence applications, the ability to apply conditional probability and Bayes' theorem is crucial. These tools enhance our understanding of data, enable predictive modeling, and promote adaptability to fluctuating conditions, playing a central role in many disciplines.

Conclusion

Conditional probability and Bayes' theorem provide valuable insights into the process of learning from data and adjusting previously held beliefs in light of new information. By mastering these concepts, you not only increase your mathematical depth but also equip yourself with powerful tools to navigate uncertainties and make well-grounded decisions in a variety of contexts.

These theories highlight the beauty of probability as a bridge between mathematical theory and practical application, which continues to evolve as part of our understanding of the world.


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Conditional Probability and Bayes' Theorem

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