Mathematics Class 12 - Probability Notes

Probability

Welcome to the chapter on Probability for Class 12. In this chapter, you will learn about the concepts of probability, types of events, random variables, probability distributions, and how to solve problems using these concepts. By the end of this chapter, you will be able to apply probability theory to real-life and mathematical situations.

Key Concepts

• Random Experiment: An experiment whose outcome cannot be predicted with certainty.
• Sample Space (S): The set of all possible outcomes of a random experiment.
• Event: A subset of the sample space.
• Probability: A measure of the likelihood that an event will occur.
• Random Variable: A variable that takes numerical values determined by the outcome of a random experiment.
• Probability Distribution: A function that gives the probability of each value of a random variable.

Classical (Theoretical) Probability

If all outcomes are equally likely, the probability of an event E is given by:

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

Example: The probability of getting a 3 when a die is thrown is 1/6.

Types of Events

• Impossible Event: Probability is 0.
• Sure Event: Probability is 1.
• Mutually Exclusive Events: Events that cannot happen at the same time.
• Exhaustive Events: All possible outcomes together.
• Independent Events: Occurrence of one does not affect the other.
• Complementary Events: The event not happening.

Addition and Multiplication Theorems

• Addition Theorem: P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
• Multiplication Theorem (Independent Events): P(A ∩ B) = P(A) × P(B)

Conditional Probability

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

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

Random Variables and Probability Distributions

• Discrete Random Variable: Takes finite or countable values.
• Probability Distribution: Assigns probabilities to each value of the random variable.
• Mean (Expected Value): E(X) = Σ[x × P(x)]
• Variance: Var(X) = E(X²) – [E(X)]²

Practice Questions

1. A card is drawn from a well-shuffled deck of 52 cards. What is the probability of drawing a king?
2. If two coins are tossed, what is the probability of getting at least one head?
3. If P(A) = 0.4, P(B) = 0.5, and P(A ∩ B) = 0.2, find P(A ∪ B).
4. A die is thrown. What is the probability of getting an even number?
5. If X is a discrete random variable with P(X=1)=0.3, P(X=2)=0.5, P(X=3)=0.2, find E(X).

Challenge Yourself

• Prove that the probability of the complement of an event A is 1 – P(A).
• If three coins are tossed, find the probability distribution of the number of heads.
• A bag contains 5 red and 7 blue balls. Two balls are drawn at random. What is the probability that both are red?

Did You Know?

• Probability is used in weather forecasting, insurance, and games of chance.
• The probability of an event always lies between 0 and 1.

Glossary

• Random Experiment: An experiment with uncertain outcome.
• Sample Space: The set of all possible outcomes.
• Event: A subset of the sample space.
• Random Variable: A variable whose value depends on the outcome of a random experiment.
• Probability Distribution: A table or formula that gives the probability of each value of a random variable.

Answers to Practice Questions

1. 4/52 = 1/13
2. 3/4
3. P(A ∪ B) = 0.4 + 0.5 – 0.2 = 0.7
4. 3/6 = 1/2
5. E(X) = 1×0.3 + 2×0.5 + 3×0.2 = 0.3 + 1.0 + 0.6 = 1.9

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Practice probability problems to master the concepts and apply them in real life!