NCERT Class 10 Maths Chapter 14: Probability Notes

Probability — A Theoretical Approach What is Probability? Probability tells us how likely an event is to happen. 🪙 Example 1: Tossing a Fair Coin A fair coin has 2 sides: Head (H) and Tail (T). Each side has an equal chance of showing up when tossed. Sample Space (S) = {H, T} Total outcomes = 2 Event Example: Getting a Head Favourable outcome = {H} Number of favourable outcomes = 1 🧮 Probability = P(Head) = 𝟏/𝟐

🎲 Example 2: Throwing a Fair Die A fair die has 6 faces numbered 1 to 6. Each number has an equal chance of appearing when the die is rolled. Sample Space (S) = {1, 2, 3, 4, 5, 6} Total outcomes = 6 Event Example: Getting an even number Favourable outcomes = {2, 4, 6} Number of favourable outcomes = 3 🧮 Probability = P(even number) = 𝟑/𝟔=𝟏/𝟐 🃏 Example 3: Drawing from a Deck of Cards A standard deck has 52 cards, divided into 4 suits: ♠️ Spades, ♥️ Hearts, ♦️ Diamonds, ♣️ Clubs. Each suit has 13 cards (A, 2–10, J, Q, K). If one card is drawn at random: All 52 cards are equally likely Sample space: 52 possible outcomes Example Event: Drawing a Queen  → Favourable outcomes: {Q♠️, Q♥️, Q♦️, Q♣️}  → Total favourable = 4 🧮 Probability = P(Queen) = 𝟒/𝟓𝟐=𝟏/𝟏𝟑 Aces Number Cards Face Cards Experiment and Probability What is an Experiment? An experiment is something we do to see what happens. It gives us results or outcomes.Two Types of Experiments:1️⃣ Deterministic Experiment These always give the same result if done in the same way. Example: A science experiment where water boils at 100°C every time. Point: The result is fixed and predictable. 2️⃣ Random (Probabilistic) Experiment These may give different results every time, even if done the same way. Example: Tossing a coin — you may get Head or Tail. You can't be sure which one will come. Point: The result is not fixed — it’s by chance. 🎲 Sample Space (S) The sample space is the list of all possible results of a random experiment. Example: Throwing a die →S = {1,2,3,4,5,6} 🎯 Event An event is a result or group of results from the sample space. Example: Getting an even number when throwing a die →E = {2,4,6} 🌟 Favourable Outcomes These are the results that match what we want. Example: If we want to get a sum of 8 when two dice are thrown: Favourable outcomes = (2,6), (3,5), (4,4), (5,3), (6,2) → 5 outcomes 🔁 Complementary Event If E is an event, then not E means the event does not happen. Formula:P(E) + P(not E) = 1 or P(E) = 1− P(not E) Example: If the chance of rain is 0.7, then Chance of no rain = 1 - 0.7 = 0.3 Assumption in This Chapter: To make things simple, we will assume that all outcomes in experiments are equally likely.Empirical (Experimental) Probability: If we repeat an experiment many times, we can find the chance of an event by:𝑃(𝐸)="Number of times event E happens" /"Total number of trials" This method works well for repeated experiments like coin tosses or dice rolls. But for some events like satellite launches or earthquakes, repeating the experiment is difficult or impossible. Theoretical (Classical) Probability: When repeating the experiment is not practical, theoretical probability is used. It assumes all possible outcomes are equally likely and calculates probability by: 𝑃(𝐸)="Number of favorable outcomes " /"Total number of possible outcomes " This gives the exact chance of an event happening without doing the experiment repeatedly.History: This definition of probability was introduced by Pierre Simon Laplace in 1795. 📝 Summary Table: Probability Basics 📐 Important Formulas: 1️⃣ Theoretical Probability: 𝑷(𝑬)="Number of times event E happens" /"Total number of trials" 2️⃣ Complementary Events: P(E) + P(not E) = 1 or P(E) = 1− P(not E)

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