## Karnataka Class 12 Commerce Maths Unit VI – Probability Complete Notes

Karnataka Class 12 Commerce Maths Unit VI – Probability : CBSE board has been offering a robust, holistic school education to students since its inception. The board first analyzes students’ learning requirements and according to that, it prepares suitable syllabus for each class. Additionally, the board designs appropriate question papers to evaluate students’ subject knowledge at the end of each academic session. Moreover, to keep students stress free during exams, the board also designs sample papers for each class. The CBSE board conducts research to get to know the current educational requirements and based on that, it chooses suitable subjects and their relevant topics. Hence, students, who are pursuing their studies under this board, get updated information and keep them prepared for any competitive exams.

### Karnataka Class 12 Commerce Maths Unit VI – Probability Complete Notes

Karnataka Class 12 Commerce Maths Unit VI – Probability : Here our team members provides you Karnataka Class 12 Commerce Maths Unit VI – Probability Complete Notes in pdf format. Here we gave direct links for you easy to download Karnataka Class 12 Commerce Maths Unit VI – Probability Complete Notes.Karnataka Class 12 Commerce Maths Unit VI – Probability topics are Conditional probability, multiplication theorem on probability. independent events, total probability, Baye’s theorem, Random variable and its probability distribution, mean and variance of random variable. Repeated independent (Bernoulli) trials and Binomial distribution. Download these Karnataka Class 12 Commerce Maths Unit VI – Probability Complete Notes and read well.

### Download here Karnataka Class 12 Commerce Maths Complete Notes In PDF Format

### Karnataka Class 12 Commerce Maths Unit VI – Probability Complete Notes

Karnataka Class 12 Commerce Maths Unit VI – Probability : **Probability** is a branch of mathematics that deals with calculating the likelihood of a given event’s occurrence, which is expressed as a number between 1 and 0. **Probability** is the measure of the likelihood that an event will occur. Probability is quantified as a number between 0 and 1, where, loosely speaking, 0 indicates impossibility and 1 indicates certainty. The higher the probability of an event, the more certain that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes (“heads” and “tails”) are both equally probable; the probability of “heads” equals the probability of “tails”; and since no other outcomes are possible, the probability of either “heads” or “tails” is 1/2 (which could also be written as 0.5 or 50%).

These concepts have been given an axiomatic mathematical formalization in probability theory, which is used widely in such areas of study as mathematics, statistics, finance, gambling, science (in particular physics), artificial intelligence/machine learning, computer science, game theory, and philosophy to, for example, draw inferences about the expected frequency of events. Probability theory is also used to describe the underlying mechanics and regularities of complex systems.

### Karnataka Class 12 Commerce Maths Unit VI – Probability Complete Notes

Karnataka Class 12 Commerce Maths Unit VI – Probability : **Probability** is the likelihood of something happening. When someone tells you the probability of something happening, they are telling you how likely that something is. When people buy lottery tickets, the probability of winning is usually stated, and sometimes, it can be something like 1/10,000,000 (or even worse). This tells you that it is not very likely that you will win.

The formula for probability tells you how many choices you have over the number of possible combinations.

## Calculating Probability

To calculate probability, you need to know how many possible options or outcomes there are and how many right combinations you have. Let’s calculate the probability of throwing dice to see how it works.

First, we know that a die has a total of 6 possible outcomes. You can roll a 1, 2, 3, 4, 5, or 6. Next, we need to know how many choices we have. Whenever you roll, you will get one of the numbers. You can’t roll and get two different numbers with one die. So, our number of choices is 1. Using our formula for probability, we get a probability of 1/6.

Our probability of rolling any of the numbers is 1/6. The probability of rolling a 2 is 1/6, of rolling a 3 is also 1/6, and so on.

Let’s try another problem. Let’s say we have a grab bag of apples and oranges. We want to find out the probability of picking an apple from the bag. One thing we need to know is the number of apples in the bag because that gives us the number of ‘correct’ choices, which is the number of our possible choices in the top part of the calculation.

We also need to know the total number of fruits in the bag, for this gives us the total number of choices we have, or the total number of options in the bottom part of the calculation. The person with the grab bag tells us there are 10 apples and 20 oranges in the bag. So, what is our probability of picking an apple? We have 10 apples, one of which we want, and a total of 30 fruits to pick from.

Our probability is 1/3 for picking an apple. If you compare this with our probability of rolling a number on a die, the probability of picking an apple from the grab bag is higher. It is more likely that we will pick an apple than that we will roll a particular number.

### Karnataka Class 12 Commerce Maths Unit VI – Probability Complete Notes

Karnataka Class 12 Commerce Maths Unit VI – Probability : Probability is a branch of mathematics that deals with calculating the likelihood of a given event’s occurrence, which is expressed as a number between 1 and 0. An event with a probability of 1 can be considered a certainty: for example, the probability of a coin toss resulting in either “heads” or “tails” is 1, because there are no other options, assuming the coin lands flat. An event with a probability of .5 can be considered to have equal odds of occurring or not occurring: for example, the probability of a coin toss resulting in “heads” is .5, because the toss is equally as likely to result in “tails.” An event with a probability of 0 can be considered an impossibility: for example, the probability that the coin will land (flat) without either side facing up is 0, because either “heads” or “tails” must be facing up. A little paradoxical, probability theory applies precise calculations to quantify uncertain measures of random events.

*plus*the number of failures of occurrences (this adds up to the total of possible outcomes):

p(a) = p(a)/[p(a) + p(b)]

Calculating probabilities in a situation like a coin toss is straightforward, because the outcomes are mutually exclusive: either one event or the other must occur. Each coin toss is an *independent* event; the outcome of one trial has no effect on subsequent ones. No matter how many consecutive times one side lands facing up, the probability that it will do so at the next toss is always .5 (50-50). The mistaken idea that a number of consecutive results (six “heads” for example) makes it more likely that the next toss will result in a “tails” is known as the *gambler’s fallacy* , one that has led to the downfall of many a bettor.

Probability theory had its start in the 17th century, when two French mathematicians, Blaise Pascal and Pierre de Fermat carried on a correspondence discussing mathematical problems dealing with games of chance. Contemporary applications of probability theory run the gamut of human inquiry, and include aspects of computer programming, astrophysics, music, weather prediction, and medicine.

### Karnataka Class 12 Commerce Maths Unit VI – Probability Complete Notes

Karnataka Class 12 Commerce Maths Unit VI – Probability : In probability theory, a **probability density function** (**PDF**), or **density** of a continuous random variable, is a function, whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a *relative likelihood* that the value of the random variable would equal that sample. In other words, while the *absolute likelihood* for a continuous random variable to take on any particular value is 0 (since there are an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer that, in any particular draw of the random variable, how much more likely it is that the random variable would equal one sample compared to the other sample.

In a more precise sense, the PDF is used to specify the probability of the random variable falling *within a particular range of values*, as opposed to taking on any one value. This probability is given by the integral of this variable’s PDF over that range—that is, it is given by the area under the density function but above the horizontal axis and between the lowest and greatest values of the range. The probability density function is non negative everywhere, and its integral over the entire space is equal to one.

The terms “*probability distribution function*” and “*probability function*” have also sometimes been used to denote the probability density function. However, this use is not standard among probabilistic and statisticians. In other sources, “probability distribution function” may be used when the probability distribution is defined as a function over general sets of values, or it may refer to the cumulative distribution function, or it may be a probability mass function (PMF) rather than the density. Further confusion of terminology exists because *density function* has also been used for what is here called the “probability mass function” (PMF). In general though, the PMF is used in the context of discrete random variables (random variables that take values on a discrete set), while PDF is used in the context of continuous random variables.

### Download here Karnataka Class 12 Commerce Maths Unit VI – Probability Complete Notes In PDF Format

### Karnataka Class 12 Commerce Maths Unit VI – Probability Complete Notes

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