Introduction to Probabilistic Modelling

🧠 1. Introduction to Probability 

 🔹 What is Probability? 

Probability is a way of measuring how likely an event is to happen. It's used when there is uncertainty about the outcome. 

 🔹 Probability Space A probability space includes: 

 Sample space (Ω): All possible outcomes of an experiment. E.g., Ω = {infected, not infected} for testing a cow for a disease. 

 Event space (𝓕): Collection of events we are interested in. E.g., event E = {infected}. 

 

Probability (P): A number between 0 and 1 assigned to each event. 

E.g., P(E) = 0.3 means a 30% chance of infection. 

 🧪 Example: Let’s say in a herd of 100 animals, 25 have a parasitic infection. P(infection) = 25/100 = 0.25 

 🔄 2. Conditional Probability 

🔹 What is it? 

Conditional probability is the chance of one event happening given that another has already occurred.

 

 📌 Formula: 𝑃 ( 𝐴 ∣ 𝐵 ) = 𝑃 ( 𝐴 ∩ 𝐵 ) 𝑃 ( 𝐵 ) P(A∣B)= P(B) P(A∩B) ​

 🧪 Example (Animal Health): Let: A = Animal is infected. B = Animal shows symptoms. 

 

Then, 𝑃 ( Infected | Symptoms ) = P(Infected and Symptoms) P(Symptoms) P(Infected | Symptoms)= P(Symptoms) P(Infected and Symptoms) ​

 This helps veterinarians determine true infection rates among symptomatic animals. 

 🧠 3. Bayes’ Rule 

Bayes’ Rule helps update our beliefs when we get new information. 

 📌 Formula: 𝑃 ( 𝐵 ∣ 𝐴 ) = 𝑃 ( 𝐴 ∣ 𝐵 ) ⋅ 𝑃 ( 𝐵 ) 𝑃 ( 𝐴 ) P(B∣A)= P(A) P(A∣B)⋅P(B) ​

 🧪 Animal Science Example: B = Disease is present A = Test result is positive Bayes’ Rule lets us calculate: "If the test is positive, what is the probability that the animal truly has the disease?" This is key in evaluating diagnostic test accuracy (Sensitivity & Specificity). 

 🔢 4. Random Variables 

A random variable assigns a number to each possible outcome of a random experiment. 

 Types: Discrete: Countable values E.g., Number of calves born per year Continuous: Any value in an interval E.g., Weight of sheep in kg 

 📌 Example: Let X = number of eggs laid by a hen in a day Then X is a discrete random variable. 

 📊 5. Probability Distributions 

These show how probabilities are distributed over different values of a random variable. 

 🔹 Discrete Distributions Bernoulli: Success/Failure (e.g., Pregnant or not) Binomial: Number of successes in n trials (e.g., infected cows in a sample of 10) 𝑃 ( 𝑋 = 𝑘 ) = ( 𝑛 𝑘 ) 𝑝 𝑘 ( 1 − 𝑝 ) 𝑛 − 𝑘 P(X=k)=( k n ​ )p k (1−p) n−k 

 🔹 Continuous Distributions Normal (Gaussian): 

Bell-shaped curve (e.g., milk yield, body weight) 

 📌 Animal Science Example: Milk yield of cows often follows a normal distribution with most cows near the average. 

 🔗 6. Conditional Distributions and Joint Probability 

🔹 Joint Probability Probability of two things happening together. 𝑃 ( 𝐴 ∩ 𝐵 ) P(A∩B) 

🔹 Conditional Distribution Given one variable’s value, what’s the distribution of the other? 

 🧪 Example (Breeding Program): X = Conception (Yes/No) Y = Type of nutrition We can compute: “What is the probability of conception given high-protein feed?” 

 🧬 7. Probabilistic vs Statistical vs Bayesian Models 

🔹 Probabilistic Model Defines how outcomes behave randomly. E.g., Each cow has a 70% chance of getting vaccinated. 

 🔹 Statistical Model Uses data to estimate unknown parameters in a family of probability models. E.g., Estimating infection rate from field data. 

 🔹 Bayesian Model Combines prior beliefs (before data) and new evidence (data) to update predictions. E.g., Prior belief: 5% infection rate After test: update this belief using Bayes' rule. 

 🧪 8. Examples from Animal Sciences Scenario Probabilistic Concept Pregnancy rate after AI Binomial distribution Predicting disease outbreak Conditional probability & Bayes' rule Estimating weight gain in goats Normal distribution Testing new vaccine effectiveness Bayesian model 

📝 Summary Probability helps us handle uncertainty in biological systems. Conditional probability and Bayes’ Rule are essential for making informed decisions. Random variables and their distributions model real-life animal science phenomena. Probabilistic, statistical, and Bayesian models are all crucial for analysis and research. 

 🎓 Suggested Activities Class discussion: Give a real-life animal science scenario and ask students which probabilistic model fits. Exercise: Calculate conditional probability from a 2x2 disease-test result table. Mini-project idea: Survey animal weights and test whether they follow a normal distribution.