Probability_Theory_Random_Variables_Events.pptx

Probability_Theory_Random_Variables_Events

1. Probability Theory Random Variables and Events

2. Probability theory is the branch of mathematics that deals with the study of random phenomena. It provides tools to measure uncertainty and predict outcomes. Introduction to Probability Theory

3. 1. To model uncertainty in real life. 2. To make predictions. 3. To analyze experiments and data. 4. To study random processes. Why Probability ?

4. Basic Terms in Probability 1. Experiment: An action with uncertain results. 2. Sample Space: Set of all possible outcomes. 3. Event: A subset of sample space. 4. Probability: Measure of chance of an event.

5. Experiment Examples 1. Tossing a coin → {Head, Tail} 2. Rolling a die → {1,2,3,4,5,6} 3. Drawing a card → {52 cards}

6. Sample Space The set of all possible outcomes of an experiment. Example: Tossing two coins → {HH, HT, TH, TT}

7. Events An event is a set of outcomes of interest. Example: Getting an even number on a die → {2,4,6}

8. Types of Events 1. Simple Event: Single outcome. 2. Compound Event: More than one outcome. 3. Certain Event: Always occurs. 4. Impossible Event: Never occurs.

9. Operations on Events 1. Union (A ∪ B): Either A or B occurs. 2. Intersection (A ∩ B): Both A and B occur. 3. Complement (A'): Event A does not occur.

10. Probability Formula P(E) = (Number of favorable outcomes) / (Total number of outcomes) Example: P(getting 3 on a die) = 1/6

11. Random Variables A random variable is a variable that takes numerical values based on outcomes of a random experiment.

12. Types of Random Variables 1. Discrete Random Variable: Takes countable values (e.g., number of heads in 3 coin tosses). 2. Continuous Random Variable: Takes infinite values (e.g., height, weight, time).

13. Discrete Random Variable Example Experiment: Tossing 2 coins. Random Variable X = Number of Heads. Possible values: {0,1,2}

14. Continuous Random Variable Example Experiment: Measuring time taken to run a race. Random Variable X = Time (seconds). Possible values: Any real number within a range.

15. Probability Distribution A probability distribution assigns probabilities to all possible values of a random variable. Example: Tossing a coin.

16. Probability Distribution Example Random Variable X = Number on a die. P(X=1)=1/6, P(X=2)=1/6, … , P(X=6)=1/6

17. Expected Value The expected value (mean) of a random variable is the average outcome if the experiment is repeated many times. Formula: E(X) = Σ[x * P(x)]

18. Variance and Standard Deviation Variance measures spread of random variable. Standard deviation is the square root of variance. Formula: Var(X) = Σ[(x - E(X))² * P(x)]

19. Applications of Probability 1. Weather prediction 2. Business risk analysis 3. Insurance 4. Games of chance 5. Machine learning & AI

20. Summary We learned about: 1. Experiments, Sample space, Events 2. Probability rules 3. Random Variables (Discrete & Continuous) 4. Probability distributions 5. Applications

21. Thank You Probability helps us understand uncertainty and make better decisions!