JEE Maths Notes: Chapter on Probability

1. Introduction to Probability

Probability is the branch of mathematics that deals with the likelihood or chance of events occurring. In this chapter, we will study the fundamentals of probability, including the definitions, properties, and various techniques used to calculate the probability of events. Probability theory is essential for solving real-world problems involving uncertainty and chance.

2. Basic Terminology in Probability

To understand probability, it's important to first familiarize ourselves with the basic terms involved. These include:

• Experiment: A process or action that results in one or more outcomes, e.g., tossing a coin.
• Sample Space (S): The set of all possible outcomes of an experiment.
• Event: A specific outcome or a set of outcomes from the sample space.
• Outcome: The result of an experiment.
• Favorable Outcomes: The outcomes that result in the occurrence of the event.

Understanding these basic terms is crucial for interpreting and calculating probability correctly.

3. Probability of an Event

The probability of an event is the ratio of the number of favorable outcomes to the total number of possible outcomes. The formula for calculating probability is:

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

The probability of an event is always a value between 0 and 1, where 0 indicates an impossible event and 1 indicates a certain event.

4. Types of Events

In probability, different types of events are considered depending on the relationship between their outcomes. The main types are:

• Independent Events: Events that do not affect each other. The occurrence of one event does not change the probability of the other. Example: Tossing two coins.
• Dependent Events: Events where the occurrence of one event affects the probability of the other event. Example: Drawing cards from a deck without replacement.
• Mutually Exclusive Events: Events that cannot happen at the same time. Example: Getting a head or a tail in a coin toss.
• Non-Mutually Exclusive Events: Events that can happen at the same time. Example: Drawing a red card and a queen from a deck.

5. Addition and Multiplication Theorems of Probability

In probability theory, the addition and multiplication theorems are fundamental in calculating the probability of combined events.

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

This formula accounts for the overlap of events A and B if they are not mutually exclusive.

P(A ∩ B) = P(A) × P(B)

For dependent events, the formula becomes:

P(A ∩ B) = P(A) × P(B|A)

Where P(B|A) is the conditional probability of event B occurring given that event A has already occurred.

6. Conditional Probability

Conditional probability refers to the probability of an event occurring given that another event has already occurred. The formula for conditional probability is:

P(A|B) = P(A ∩ B) / P(B)

Where P(A|B) is the probability of event A occurring given that event B has occurred, and P(A ∩ B) is the probability of both events A and B occurring simultaneously.

7. Bayes’ Theorem

Bayes’ Theorem is a powerful tool in probability theory used to update the probability of an event based on new evidence. It is particularly useful in problems where conditional probability is involved. The formula for Bayes' Theorem is:

P(A|B) = [P(B|A) × P(A)] / P(B)

This formula allows us to reverse conditional probabilities and is widely used in fields such as statistics, machine learning, and medical diagnostics.

8. Random Variables and Probability Distributions

A random variable is a variable whose value is determined by the outcome of a random experiment. Probability distributions describe the probability of each possible value of a random variable. There are two main types of probability distributions:

9. Expected Value and Variance

In probability, the expected value (also called the mean) is the long-term average value of a random variable. The formula for the expected value of a discrete random variable X is:

E(X) = Σ [x × P(x)]

Where x represents the possible values of the random variable and P(x) is the probability of each value.

The variance of a random variable measures how much the values deviate from the expected value. The formula for variance is:

Var(X) = Σ [(x - E(X))² × P(x)]

10. Problems and Examples

To master the concept of probability, solving problems and examples is essential. Some practice problems include:

These problems cover a range of difficulties, from basic probability to more advanced applications of conditional probability and Bayes’ Theorem.

11. Applications of Probability

Probability is widely used in various fields such as:

12. Summary and Key Takeaways

In conclusion, probability is a fundamental concept in mathematics with vast applications in various fields. The key concepts covered in this chapter include basic probability, types of events, theorems of probability, conditional probability, Bayes' Theorem, random variables, and probability distributions. Mastery of these concepts is crucial for solving probability problems in JEE and other competitive exams.

• Addition Theorem: This theorem is used when calculating the probability of the union of two events. For two events A and B, the formula is:
• Multiplication Theorem: This theorem is used to find the probability of the intersection of two events. For two independent events A and B, the formula is:
  • Discrete Probability Distributions: These distributions describe random variables that take on a finite or countably infinite number of values. Example: The number of heads when tossing three coins.
  • Continuous Probability Distributions: These distributions describe random variables that can take on any value within a given range. Example: The height of a person.
  • Example 1: Find the probability of drawing a red card from a deck of cards.
  • Example 2: Calculate the probability of getting two heads when flipping two coins.
  • Example 3: Using Bayes' Theorem to determine the probability of a person being sick given a positive test result.
  • Statistics: Probability forms the foundation of statistics, helping to analyze data and make predictions based on uncertainty.
  • Finance: In finance, probability helps in assessing risk and return on investments.
  • Computer Science: Probability is used in algorithms, data structures, and artificial intelligence for decision-making under uncertainty.
  • Games and Gambling: Probability is essential in calculating odds in games of chance.