Quick Revision Notes: Class 11 Mathematics

Chapter: Probability (प्रायिकता)

### 1. Important Terms & Definitions

• Random Experiment (यादृच्छिक प्रयोग): An experiment whose outcomes cannot be predicted with certainty in advance, even though all possible outcomes are known.
• Sample Space (प्रतिदर्श समष्टि - $S$): The set of all possible outcomes of a random experiment.
  • Example: Rolling a die: $S = {1, 2, 3, 4, 5, 6}$
• Event (घटना - $E$): A subset of the sample space ($E \subset S$).
• Elementary Event (प्रारंभिक घटना): An event having only a single outcome of the sample space.
• Compound Event (संयुक्त घटना): An event having more than one outcome.
• Sure Event (निश्चित घटना): An event that always occurs (equal to sample space $S$).
• Impossible Event (असंभव घटना): An event that cannot occur (equal to empty set $\phi$).

### 2. Types of Events (घटनाओं کے प्रकार)

• Complementary Event (पूरक घटना - $E'$ or $\bar{E}$): "Not $E$". The event representing all outcomes that are not in $E$.
  • $\bar{E} = S - E$
• Mutually Exclusive Events (परस्पर अपवर्जी घटनाएँ): Two events $A$ and $B$ are mutually exclusive if they cannot occur simultaneously.
  • $A \cap B = \phi \implies P(A \cap B) = 0$
• Exhaustive Events (निःशेष घटनाएँ): A set of events $A_1, A_2, \dots, A_n$ is exhaustive if their union is the entire sample space $S$.
  • $A_1 \cup A_2 \cup \dots \cup A_n = S \implies P(A_1 \cup A_2 \cup \dots \cup A_n) = 1$

### 3. Classical Definition of Probability (प्रायिकता की शास्त्रीय परिभाषा)

For a finite sample space $S$ with equally likely outcomes, the probability of an event $E$ is given by:

$$P(E) = \frac{\text{Number of favourable outcomes to } E}{\text{Total number of outcomes in } S} = \frac{n(E)}{n(S)}$$

Key Rules:

1. $0 \le P(E) \le 1$
2. $P(S) = 1$ (Sure Event)
3. $P(\phi) = 0$ (Impossible Event)
4. $P(\bar{E}) = 1 - P(E)$ or $P(E) + P(\bar{E}) = 1$

### 4. Addition Theorems of Probability (योग प्रमेय)

• General Addition Rule (सामान्य योग नियम): For any two events $A$ and $B$: $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

• For Mutually Exclusive Events ($A \cap B = \phi$): $$P(A \cup B) = P(A) + P(B)$$

• For Three Events ($A, B, C$): $$P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(A \cap C) + P(A \cap B \cap C)$$

### 5. Useful Relations (महत्वपूर्ण संबंध)

• $P(A \text{ or } B) = P(A \cup B)$
• $P(A \text{ and } B) = P(A \cap B)$
• Difference Event ($A - B$ or $A \text{ but not } B$): $$P(A - B) = P(A) - P(A \cap B)$$
• De Morgan's Laws:
  • $\overline{A \cup B} = \bar{A} \cap \bar{B} \implies P(\overline{A \cup B}) = 1 - P(A \cup B)$
  • $\overline{A \cap B} = \bar{A} \cup \bar{B} \implies P(\overline{A \cap B}) = 1 - P(A \cap B)$
• Probability of occurrence of exactly one of the events $A$ or $B$: $$P(A \text{ only}) + P(B \text{ only}) = P(A) + P(B) - 2P(A \cap B)$$

### 6. Important Tips for Problem Solving (परीक्षा उपयोगी टिप्स)

1. At least one (कम से कम एक): $P(\text{At least one } A) = 1 - P(\text{None of them})$.
2. Cards Problems (ताश के पत्तों पर आधारित):
   • Total cards = 52
   • Red cards = 26 (Hearts ♥ 13, Diamonds ♦ 13)
   • Black cards = 26 (Spades ♠ 13, Clubs ♣ 13)
   • Face cards (तस्वीर वाले पत्ते) = 12 (4 Kings, 4 Queens, 4 Jacks)
3. Coins Problems (सिक्के):
   • For $n$ coins tossed, $n(S) = 2^n$.
4. Dice Problems (पासे):
   • For $n$ dice rolled, $n(S) = 6^n$.