ABI Bioinformatics Guide 2024
  • INTRODUCTION
    • How to use the guide
  • MOLECULAR BIOLOGY
    • The Cell
      • Cells and Their Organelles
      • Cell Specialisation
      • Quiz 1
    • Biological Molecules
      • Carbohydrates
      • Lipids
      • Nucleic Acids (DNA and RNA)
      • Quiz 2
      • Proteins
      • Catalysis of Biological Reactions
      • Quiz 3
    • Information Flow in the Cell
      • DNA Replication
      • Gene Expression: Transcription
      • Gene Expression: RNA Processing
      • Quiz 4
      • Chromatin and Chromosomes
      • Regulation of Gene Expression
      • Quiz 5
      • The Genetic Code
      • Gene Expression: Translation
    • Cell Cycle and Cell Division
      • Quiz 6
    • Mutations and Variations
      • Point mutations
      • Genotype-Phenotype Interactions
      • Quiz 7
  • PROGRAMMING
    • Python for Genomics
    • R programming (optional)
  • STATISTICS: THEORY
    • Introduction to Probability
      • Conditional Probability
      • Independent Events
    • Random Variables
      • Independent, Dependent and Controlled Variables
    • Data distribution PMF, PDF, CDF
    • Mean, Variance of a Random Variable
    • Some Common Distributions
    • Exploratory Statistics: Mean, Median, Quantiles, Variance/SD
    • Data Visualization
    • Confidence Intervals
    • Comparison tests, p-value, z-score
    • Multiple test correction: Bonferroni, FDR
    • Regression & Correlation
    • Dimentionality Reduction
      • PCA (Principal Component Analysis)
      • t-SNE (t-Distributed Stochastic Neighbor Embedding)
      • UMAP (Uniform Manifold Approximation and Projection)
    • QUIZ
  • STATISTICS & PROGRAMMING
  • BIOINFORMATICS ALGORITHMS
    • Introduction
    • DNA strings and sequencing file formats
    • Read alignment: exact matching
    • Indexing before alignment
    • Read alignment: approximate matching
    • Global and local alignment
  • NGS DATA ANALYSIS & FUNCTIONAL GENOMICS
    • Experimental Techniques
      • Polymerase Chain Reaction
      • Sanger (first generation) Sequencing Technologies
      • Next (second) Generation Sequencing technologies
      • The third generation of sequencing technologies
    • The Linux Command-line
      • Connecting to the Server
      • The Linux Command-Line For Beginners
      • The Bash Terminal
    • File formats, alignment, and genomic features
      • FASTA & FASTQ file formats
      • Basic Unix Commands for Genomics
      • Sequences and Genomic Features Part 1
      • Sequences and Genomic Features Part 2: SAMtools
      • Sequences and Genomic Features Part 3: BEDtools
    • Genetic variations & variant calling
      • Genomic Variations
      • Alignment and variant detection: Practical
      • Integrative Genomics Viewer
      • Variant Calling with GATK
    • RNA Sequencing & Gene expression
      • Gene expression and how we measure it
      • Gene expression quantification and normalization
      • Explorative analysis of gene expression
      • Differential expression analysis with DESeq2
      • Functional enrichment analysis
    • Single-cell Sequencing and Data Analysis
      • scRNA-seq Data Analysis Workflow
      • scRNA-seq Data Visualization Methods
  • FINAL REMARKS
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  1. STATISTICS: THEORY
  2. Introduction to Probability

Independent Events

PreviousConditional ProbabilityNextRandom Variables

Last updated 10 months ago

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In probability theory, two events are considered independent if the occurrence of one event does not affect the probability of the occurrence of the other event. In other words, knowing that one event has occurred does not change the likelihood of the other event occurring.

For two events A and B, they are independent if and only if the probability of both events occurring together (the intersection of A and B) is equal to the product of their individual probabilities. Mathematically, this is expressed as:

P(A∩B)=P(A)×P(B)P(A \cap B) = P(A) \times P(B)P(A∩B)=P(A)×P(B)

If this equation holds true, then AAA and BBB are independent events.

For example, consider the events of flipping a coin and rolling a die. Let event AAA be getting heads on the coin flip, and event BBB be rolling a 4 on the die. The probability of getting heads on the coin is 0.50.50.5, and the probability of rolling a 4 on the die is 16\frac{1}{6}61​​. Since these two events do not influence each other, they are independent. Therefore, the probability of both events happening together is:

P(A∩B)=P(A)×P(B)=0.5×16=112P(A \cap B) = P(A) \times P(B) = 0.5 \times \frac{1}{6} = \frac{1}{12}P(A∩B)=P(A)×P(B)=0.5×61​=121​

This confirms that the events are independent because the probability of their intersection is equal to the product of their individual probabilities.