Quick Revision Notes & Formula Sheet

Class: 10th Mathematics
Chapter: Probability (प्रायिकता)

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### 1. Important Definitions (महत्वपूर्ण परिभाषाएँ)

• Probability (प्रायिकता): A quantitative measure of the likelihood or chance of occurrence of an event. (किसी घटना के घटने की संभावना का संख्यात्मक माप।)
• Experiment (प्रयोग): An action or operation that results in one or several outcomes.
• Sample Space (प्रतिदर्श समष्टि): The set of all possible outcomes of a random experiment.
• Event (घटना): A subset of the sample space. An event is associated with certain outcomes of an experiment.
• Elementary Event (प्रारंभिक घटना): An event that has only one outcome of the sample space. (वह घटना जिसका केवल एक ही परिणाम हो।)
• Complementary Event (पूरक घटना): The event representing "not E", denoted by E' or P(not E). It represents all outcomes that are not in event E.

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### 2. Key Formulas & Concepts (मुख्य सूत्र और अवधारणाएँ)

1. Basic Probability Formula

The probability of occurrence of any event $E$, denoted by $P(E)$, is defined as: P(E) = (Number of outcomes favourable to E) / (Total number of possible outcomes of the experiment)

In short: P(E) = n(E) / n(S) Where $n(E)$ is the number of favorable outcomes and $n(S)$ is the total number of outcomes.

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2. Relationship Between an Event and its Complement

For any event $E$: P(E) + P(not E) = 1 OR P(E') = 1 - P(E) OR P(E) = 1 - P(not E)

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3. Range of Probability (प्रायिकता का परिसर)

• The probability of an event always lies between $0$ and $1$ (inclusive).
• 0 <= P(E) <= 1
• Expressed in percentage: $0%$ to $100%$.

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4. Types of Special Events

• Sure Event / Certain Event (िश्चित घटना): An event that is sure to happen. Its probability is always $1$.
• Impossible Event (असम्भव घटना): An event that cannot happen. Its probability is always $0$.
• Sum of Elementary Events: The sum of the probabilities of all the elementary events of an experiment is $1$. P(E1) + P(E2) + ... + P(En) = 1

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### 3. Important Standard Problems & Shortcuts (महत्वपूर्ण मानक प्रश्न)

A. Coin Based Problems (सिक्के पर आधारित प्रश्न)

• Single Coin: Total outcomes $S = {H, T}$ -> Total = $2^1 = 2$
• Two Coins: Total outcomes $S = {HH, HT, TH, TT}$ -> Total = $2^2 = 4$
• Three Coins: Total outcomes $S = 2^3 = 8$

B. Die / Dice Based Problems (पासे पर आधारित प्रश्न)

• Single Die: Total outcomes $S = {1, 2, 3, 4, 5, 6}$ -> Total = $6^1 = 6$
• Two Dice: Total outcomes $6^2 = 36$
• Key terms:
  • Prime Number (अभाज्य संख्या): $2, 3, 5$ (Total = 3)
  • Even Number (सम संख्या): $2, 4, 6$ (Total = 3)
  • Odd Number (विषम संख्या): $1, 3, 5$ (Total = 3)

C. Playing Cards Based Problems (ताश की गड्डी पर आधारित प्रश्न)

• Total Cards: $52$ cards divided into 4 suits (Spades, Clubs, Hearts, Diamonds).
• Red Cards (लाल पत्ते): $26$ (13 Hearts + 13 Diamonds)
• Black Cards (काले पत्ते): $26$ (13 Spades + 13 Clubs)
• Face Cards (तस्वीर वाले पत्ते): $12$ (4 Kings, 4 Queens, 4 Jacks)
• Honor / Ace Cards (इक्के): $4$ Aces.

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All the Best for your MP Board Class 10 Examinations!