Probability Distribution Notes: Probability is a fundamental aspect of mathematics that helps us understand and quantify uncertainty. Mastery of this subject is essential for students, as it has practical applications in various fields.
This article presents detailed notes on Probability, designed specifically for students up to class 12. The notes have been prepared by taking the guidance of subject experts and referring to NCERT books, helping students grasp the essentials of this important mathematical field. Here, you will explore the basics of probability, including key terms, types of events, and important formulas. Additionally, you will learn about conditional probability, random variables, probability distributions, and real-life applications of probability. Whether you're preparing for exams or want a deeper understanding, these notes offer valuable insights and practical examples to enhance your knowledge of probability.
Probability Definition
Probability is a branch of mathematics that deals with the likelihood or chance of different outcomes.
Basic Concepts Related to Probability
1. Experiment
Any action or process with uncertain results, such as tossing a coin or rolling a dice is termed as an experiment in probability.
2. Outcome
A possible result of an experiment is called outcome. For example, getting heads or tails in a coin toss.
3. Sample Space
The set of all possible outcomes of an experiment is called sample space. For example, {Heads, Tails} on tossing a coin.
4. Event
A subset of the sample space of an experiment is called an event.
For example, getting an even number E1 = {2, 4, 6}when rolling a die is an event. E1 = {2, 4, 6} is a subset of {1, 2, 3, 4, 5, 6} which is the sample space of rolling a die.
5. Probability of an Event (P)
A measure of how likely an event is to occur is called probability of an event. It is calculated by:
6. Types of Events
- Simple Event: It is an event with a single outcome. For example, getting a 3 when rolling a die.
- Compound Event: It is an event with more than one outcome. For example, getting an even number when rolling a die.
- Certain Event: It is an event that is sure to happen. It’s probability = 1.
- Impossible Event: It is an event that cannot happen. It’s probability = 0.
- Complementary Event: The complement of any event E is the event [not E] or denoted as E', i.e. the event that E does not occur. The event E and its complement E' are mutually exclusive and exhaustive. For example, if E is getting an even number, E' is getting an odd number).
Formula for Probability
The probability of an event is expressed as the ratio of the number of favourable outcomes to the total number of outcomes of the event.
Examples of Probability
1. What is the probability that a card taken from a standard deck, is an Ace?
Solution:
Total number of cards a standard pack contains = 52
Number of Ace cards in a deck of cards = 4
So, the number of favourable outcomes = 4
Using the probability formula, we have:
Probability of selecting an ace, P(Ace) = (Number of favourable outcomes) / (Total number of outcomes)
P(Ace) = 4/52 = 1/13
So, the probability of getting an ace is 1/13.
2. Calculate the probability of getting an odd number if a dice is rolled.
Solution:
For rolling a dice the sample space (S) = {1, 2, 3, 4, 5, 6}
∴Total number of outcomes, n(S) = 6
Let “E” be the event of getting an odd number, then
E = {1, 3, 5}
∴Number of favourable outcomes, n(E) = 3
So, the Probability of getting an odd number is, P(E) = (Number of favorable outcomes )/(Total number of outcomes)
= n(E)/n(S) = 3/6 = 1/2
Thus, the probability of getting an odd number on rolling a dice is 1/2.
Rules of Probability
Addition Rule
- For mutually exclusive events A and B: P(A or B)=P(A)+P(B)
- For non-mutually exclusive events: P(A or B)=P(A)+P(B)−P(A∩B)
Multiplication Rule
- For independent events A and B: P(A and B)=P(A)×P(B)
- For dependent events: P(A and B)=P(A)×P(B∣A), where P(B∣A) is the conditional probability of B given A.
Conditional Probability
- Definition: The probability of an event occurring given that another event has already occurred.
- Formula: P(A∣B)=P(A∩B)/P(B)
- Example: In a deck of cards, the probability of drawing a king given that a face card has been drawn will be the conditional probability.
Random Variables and Probability Distributions
- Random Variable: It is a real-valued function that assigns a numerical value to each possible outcome of a random experiment. For example, in the case of the tossing of an unbiased coin, if there are 3 trials, then the number of times a ‘head’ appears can be a random variable.
- Probability Distribution: A probability distribution is a mathematical function that describes the probability of different possible values of a variable.
Probability in Real Life
Below are some of the important real life applications of probability:
- Weather Forecasting: Predicting the likelihood of weather conditions.
- Insurance: Calculating premiums based on risk.
- Games and Sports: Determining odds and strategies.
- Finance: Assessing investment risks and returns.
Common Misconception in Probability
- Gambler's Fallacy: The gambler's fallacy is the belief that, if an event (whose occurrences are independent and identically distributed) has occurred less frequently than expected, it is more likely to happen again in the future.
These notes provide a comprehensive overview of probability, suitable for students up to class 12, and are designed to build a solid foundation in understanding and applying probability concepts.
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