Probability Notes: Class 12 Mathematics Chapter 13
Class 12 Mathematics Chapter 13 Probability covers conditional probability, the multiplication theorem, independent events, Bayes' theorem, and probability distributions of random variables, including the Binomial distribution.
Overview of Topics & Subtopics (Click to Jump)
1. Conditional Probability and Multiplication Theorem
Short Note:
• Conditional probability P(A|B) represents the probability of event A occurring given that event B has already occurred, and is calculated as P(A∩B)/P(B), provided P(B) ≠ 0.
• The multiplication theorem states that P(A∩B) = P(A) · P(B|A) = P(B) · P(A|B), which is used to find the probability of two events occurring together.
2. Independent Events and Bayes' Theorem
Short Note:
• Two events A and B are said to be independent if the occurrence of one does not affect the probability of the other, expressed mathematically as P(A∩B) = P(A) · P(B).
• Bayes' theorem is used to find the probability of a particular cause, given that a certain observed effect has occurred, by combining prior probabilities with conditional probabilities.
3. Random Variables and the Binomial Distribution
Short Note:
• A random variable assigns a numerical value to each outcome of a random experiment, and its probability distribution lists all possible values along with their associated probabilities.
• The Binomial distribution applies to a fixed number of independent trials, each with only two possible outcomes (success or failure), and is used to calculate the probability of obtaining a specific number of successes.
NCERT Textbook Exercise Questions
Define conditional probability and write its formula.
Answer: Conditional probability refers to the probability of an event A occurring, given that another event B has already occurred, and is denoted as P(A|B). It is calculated using the formula P(A|B) = P(A∩B)/P(B), which is defined only when P(B) is not equal to zero.
When are two events said to be independent? Give an example.
Answer: Two events A and B are said to be independent if the occurrence or non-occurrence of one event has no effect on the probability of the other event occurring, mathematically expressed as P(A∩B) = P(A) · P(B). For example, when tossing a fair coin and rolling a fair die simultaneously, the outcome of the coin toss and the outcome of the die roll are independent of each other.
State Bayes' theorem and explain its practical use.
Answer: Bayes' theorem provides a way to calculate the probability of a particular cause or event Ei, given that a certain observed outcome A has occurred, using prior probabilities combined with conditional probabilities. It is particularly useful in situations such as medical diagnosis or quality control, where the probability of a specific cause needs to be updated after observing new evidence.
What conditions must be satisfied for a random experiment to follow a Binomial distribution?
Answer: For a random experiment to follow a Binomial distribution, it must consist of a fixed number of independent trials, each trial must have only two possible outcomes (commonly called success and failure), and the probability of success must remain constant across all trials. Under these conditions, the probability of obtaining exactly k successes in n trials can be calculated using the binomial probability formula.
