Content Strategy Manager
The NCERT Solutions for class 12 mathematics chapter 13: Probability are provided in the article below. Probability is the branch of mathematics that states how likely an event is to occur in mathematical context. There are events whose outcomes can not be predicted with full certainty. In this case, the best we can say is how likely they are to happen by using the idea of probability.
The chapter, Probability itself makes up to a whole unit, Unit Six- Probability, that carries 8 marks of the total 80 marks.
Download: NCERT Solutions for Class 12 Mathematics Chapter 13 pdf
Class 12 Maths NCERT Solutions Chapter 13 Probability
Important Topics in Class 12 Mathematics Chapter 13 Probability
Important topics covered in Class 12 Mathematics Chapter 13 Probability are:
Conditional Probability
Conditional probability is the likelihood of an event or outcome occurring, based on occurrence of a previous event or outcome. Conditional probability can be calculated by multiplying probability of the preceding event by the updated probability of the succeeding, or conditional, event.
| Examples of Conditional Probability:
|
Multiplication Theorem on Probability
Multiplication theorem of probability defines the condition between two given events. For two events, X and Y associated with a sample space S, X∩Y denotes the events in which both events have occurred. This is also known as the multiplication theorem in probability. The probabilities of the two given events are multiplied to give the probability of those events occurring simultaneously.
Taking the above example of throwing of two dice, the possible outcomes are:
S = {(x, y): x, y = 1, 2, 3, 4, 5, 6}
Independent Events
An independent event has no connection to another event’s chances of happening (or not happening). When two events are independent, one event does not influence the probability of another event.
| Simple examples of independent events:
|
Bayes’ Theorem
In probability, the Bayes’ theorem (also known as the Bayes’ rule) is a mathematical formula which determines the conditional probability of events. Bayes’ theorem describes the probability of an event based on prior knowledge of the conditions that might be relevant to the event.
The Bayes’ theorem is expressed in the following formula:
P(A|B) = \(\frac{P(B|A)P(A)}{P(B)}\)
Here:
- P(A|B)– Probability of event A occurring, given event B has occurred
- P(B|A)– Probability of event B occurring, given event A has occurred
- P(A)– Probability of event A
- P(B)– Probability of event B
Random Variables and its Probability Distributions
Probability distribution for a random variable describes how probabilities are distributed over the values of a random variable. For a discrete random variable, x, probability distribution is defined by a probability mass function, denoted by f(x).
- Mean of a random variable
Mean of the discrete random variable X is also called the expected value of X. Notationally, expected value of X is denoted by E(X). Here is the formula for calculating the mean of a discrete random variable:
\(E(X) = \mu_x = \Sigma [x_i \times P(x_i)]\)
- Variance of a random variable
Variance of a random variable is referred as the expected value of the squared deviation from the mean of \(X, \mu=E[X]\). This definition encompasses random variables that are generated by processes that are discrete, continuous, neither, or mixed.
Bernoulli Trials and Binomial Distribution
Bernoulli distribution is the discrete probability distribution of a random variable that takes a binary, boolean output: 1 with probability p, and 0 with probability (1-p).
The idea is that, whenever an experiment is being conducted, which might lead either to a success or to a failure, success can be associated (labeled with 1) a probability p, while failure (labeled with 0) will have probability (1-p).
A random experiment whose outcomes are only of two types, say success S and failure F, is a Bernoulli trial. The probability of success is taken as p while that of failure is q = 1 − p.
For example: A random experiment of items in a sale are either sold or not sold. A manufactured item can be defective or non-defective. An egg is either boiled or not boiled.
Exercise Solutions of Class 12 Maths Chapter 13 Probability
Also check Exercise Solutions of Class 12 Maths Chapter 13 Probability
- NCERT Solutions for Class 12 Chapter 13 Probability Exercise 13.1
- NCERT Solutions for Class 12 Chapter 13 Probability Exercise 13.2
- NCERT Solutions for Class 12 Chapter 13 Probability Exercise 13.3
- NCERT Solutions for Class 12 Chapter 13 Probability Exercise 13.4
- NCERT Solutions for Class 12 Chapter 13 Probability Exercise 13.5
- NCERT Solutions for Class 12 Chapter 13 Probability Miscellaneous Exercises
Also Read:
Check-Out:
Comments