🎲 Probability (Class 12 NCERT) – Summary

📌 1. Introduction

Probability is the branch of mathematics that deals with uncertainty and randomness 🎯. It helps in predicting the chances of different outcomes in real-life situations.

🔄 2. Random Experiment

A random experiment is an experiment whose outcome cannot be predicted with certainty 🎲. Examples include tossing a coin or rolling a die.

📦 3. Sample Space

The sample space (S) is the set of all possible outcomes of a random experiment. Each outcome is called a sample point.

🎯 4. Event

An event is a subset of the sample space.

• ✅ Simple Event – one outcome
• 🔗 Compound Event – more than one outcome
• ❌ Impossible Event – no outcome
• ✔️ Sure Event – all outcomes

📊 5. Classical Definition of Probability

If all outcomes are equally likely:

P(E) = Number of favorable outcomes / Total number of outcomes

📐 6. Axiomatic Approach to Probability

• 📌 P(E) ≥ 0
• 📌 P(S) = 1
• 📌 For mutually exclusive events: P(E₁ ∪ E₂) = P(E₁) + P(E₂)

🧠 7. Properties of Probability

• 🔢 0 ≤ P(E) ≤ 1
• ⭕ P(∅) = 0
• 🔁 P(E') = 1 − P(E)

🚫 8. Mutually Exclusive Events

Events that cannot occur at the same time are called mutually exclusive.

🔄 9. Complementary Events

The complement of an event E represents the non-occurrence of E.

📎 10. Conditional Probability

The probability of event A occurring given that event B has occurred is:

P(A | B) = P(A ∩ B) / P(B), where P(B) ≠ 0

✖️ 11. Multiplication Theorem of Probability

For two events A and B:

P(A ∩ B) = P(A) × P(B | A)

🔗 12. Independent Events

Two events are independent if one does not affect the occurrence of the other.

P(A ∩ B) = P(A) × P(B)

🧩 13. Bayes’ Theorem

Bayes’ theorem helps in finding conditional probabilities 🔍.

P(A₁ | B) = [P(A₁) × P(B | A₁)] / Σ[P(Aᵢ) × P(B | Aᵢ)]

📈 14. Random Variable

A random variable assigns numerical values to outcomes of a random experiment.

📑 15. Probability Distribution

It shows how probabilities are assigned to values of a random variable.

• 📌 Total probability =