Class 11 Ganit Ch-17 प्रायिकता Probability

Probability (प्रायिकता) — Complete Master Hierarchy (Basic To Advanced)

Random Experiment (यादृच्छिक प्रयोग)

• Definition (परिभाषा)
  • Deterministic vs Random Experiment
  • Conditions of Randomness
  • Uncertainty Principle
• Types (प्रकार)
  • Simple Experiment
  • Compound Experiment
  • Finite Experiment
  • Infinite Experiment
• Characteristics (विशेषताएँ)
  • Repeatability
  • Unpredictability
  • Well-defined Outcomes

Trial and Outcomes (प्रयोग और परिणाम)

• Trial (प्रयोग)
  • Single Trial
  • Multiple Trials
• Outcomes (परिणाम)
  • Possible Outcomes
  • Favorable Outcomes

Sample Space (नमूना स्थान)

• Definition
  • Complete Set of Outcomes
• Representation
  • Listing Method
  • Set Builder Method
  • Tree Diagram
• Types
  • Finite Sample Space
  • Infinite Sample Space
• Size
  • Cardinal Number n(S)

Events (घटनाएँ)

• Definition
  • Subset of Sample Space
• Simple Event (सरल घटना)
  • Single Outcome Event
  • Elementary Event
• Compound Event (संयुक्त घटना)
  • Multiple Outcomes Event
  • Union of Events
• Types of Events (घटनाओं के प्रकार)
  • Impossible Event
  • Sure Event
  • Complementary Event
  • Mutually Exclusive Events
  • Non-Mutually Exclusive Events
  • Exhaustive Events
  • Equally Likely Events
  • Independent Events
  • Dependent Events

Algebra of Events (घटनाओं का बीजगणित)

• Operations on Events
  • Union (A ∪ B)
  • Intersection (A ∩ B)
  • Complement (A’)
• Laws of Events
  • Commutative Law
  • Associative Law
  • Distributive Law

Probability of an Event (घटना की प्रायिकता)

• Definition
  • Ratio of Favorable Outcomes to Total Outcomes
• Types of Probability
  • Classical Probability
  • Empirical Probability
• Formula
  P(A)=n(A)n(S)P(A)=\frac{n(A)}{n(S)}P(A)=n(S)n(A)​
• Range
  • 0 ≤ P(A) ≤ 1
• Special Cases
  • P(∅) = 0
  • P(S) = 1
• Properties
  • Additivity Rule
  • Complement Rule

Addition Rule of Probability

• Mutually Exclusive Events
  $P(A\cup B)=P(A)+P(B)$
• Non-Mutually Exclusive Events

$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

Complementary Probability

• Rule
  P(A′)=1−P(A)P(A’)=1-P(A)P(A′)=1−P(A)

Conditional Probability (सशर्त प्रायिकता)

• Definition
  • Probability of A given B
• Formula

P(A∣B)=P(A∩B)P(B)P(A|B)=\frac{P(A\cap B)}{P(B)}P(A∣B)=P(B)P(A∩B)​

Multiplication Rule of Probability

• For Independent Events

$$P(A\cap B)=P(A)\cdot P(B)$$

• For Dependent Events

$$P(A\cap B)=P(A)\cdot P(B|A)$$

Equally Likely Outcomes (समान संभाव्य परिणाम)

• Definition
  • All Outcomes Have Equal Chance
• Conditions
  • Symmetry
  • Fair Experiment
• Applications
  • Coin Toss
  • Dice
  • Cards

Odds (अनुपात आधारित प्रायिकता)

• Odds in Favour
  • Favorable : Unfavorable
• Odds Against
  • Unfavorable : Favorable

Empirical Probability (प्रायोगिक प्रायिकता)

• Definition
  • Based on Actual Experiments
• Formula
  P(A)=Number of times event A occursTotal number of trialsP(A)=\frac{\text{Number of times event A occurs}}{\text{Total number of trials}}P(A)=Total number of trialsNumber of times event A occurs​

Advanced Concepts (उन्नत विषय)

• Bayes’ Theorem
  P(Ai∣B)=P(B∣Ai)P(Ai)∑P(B∣Aj)P(Aj)P(A_i|B)=\frac{P(B|A_i)P(A_i)}{\sum P(B|A_j)P(A_j)}P(Ai​∣B)=∑P(B∣Aj​)P(Aj​)P(B∣Ai​)P(Ai​)​
• Random Variables (यादृच्छिक चर)
  • Discrete Random Variable
  • Continuous Random Variable
• Probability Distribution
  • Probability Mass Function (PMF)
  • Probability Density Function (PDF)
• Expectation (अपेक्षित मान)
  • Mean (E[X])
• Variance and Standard Deviation
  • Variance
  • Standard Deviation