Probability defines the likelihood of the occurrence of an event. There are many real-life situations in which people may have to predict the outcome of an event. They may be sure or not sure of the results of an event. In such cases, people say that there is a Probability that this event will occur or not occur. In general, probability has numerous applications in games, in business to make probability-based predictions, and in this new area of artificial intelligence. The Probability of an event can be calculated using the Probability Formulas by simply dividing the favourable number of outcomes by the total number of possible outcomes. The chance of an event occurring can range between 0 and 1 because the favourable number of outcomes can never exceed the entire number of outcomes. Additionally, the proportion of positive outcomes cannot be negative.
What is Probability?
The probability of an event is defined as the ratio of favourable outcomes to all possible outcomes.The number of positive results for an experiment with n outcomes can be represented by the symbol x. The Probability Formulas of an event can be calculated using the following Probability Formulas.
Probability Formulas(Event) = Favorable Outcomes/Total Outcomes = x/n
In order to better understand the chapter on Probability, students need to make straightforward applications. They need to forecast if it will rain or not. Either “Yes” or “No” is the appropriate response to this query. It is possible that it will rain or not.Here, the Probability Formulas can be used. Using probability, one may forecast the results of a coin toss, a roll of the dice, or a card draw.
Both theoretical Probability Formulas and experimental Probability Formulas are subcategories of probability.
Terminology of Probability Theory
The Probability Formulas terminology listed below aids in a better understanding of probability concepts.
Experiment: A trial or procedure carried out to generate a result is referred to as an “experiment.”
Sample Space: A sample space is the collection of all potential results of an experiment. Tossing a coin, for instance, has two possible outcomes: heads or tails.
Favorable Consequence: An occurrence is deemed to have produced the desired outcome or an anticipated event if it did so. For instance, if a person rolls two dice and gets the sum of the two numbers as 4, the possible or favourable possibilities are (1,3), (2,2), and (3,1).
Trial: To conduct a trial is to conduct a random experiment.
Random Experiment: A random experiment is a study that has a predetermined set of results. Tossing a coin, for instance, gives people the ption of getting either the head or the tail, but they never know for sure which one will come up.
opEvent: An event is the whole variety of results from a random experiment.
Equally Likely Events: Equally likely events are those that have the same likelihood of occurring. One event’s consequence is unrelated to another’s. For instance, there are equal possibilities of receiving either a head or a tail when a person flips a coin.
Exhaustive Events: An exhaustive event is one in which the set of all experiment results equals the sample space.
Mutually Exclusive Events: Mutually exclusive events are those that cannot take place at the same time. The weather, for instance, could be hot or cold. People cannot experience the exact same weather at the same time.
The area of Mathematics known as Probability Formulas is concerned with the numerical representation of how likely it is for an event to occur or for a specific claim to be true. Any event’s Probability ranges from 0 to 1. 1 denotes assurance that the event will occur, while 0 denotes the impossibility of the event occurring.
The Probability that an event will occur is determined by the Probability Formulas. It is the ratio of positive outcomes to total outcomes. The Probability Formulas is represented as follows:
Probability Formulas = Number of favourable outcomes to A/ total number of possible outcomes where,
The probability of an event “B” is P(B).
The number of positive outcomes of an event “B” is denoted by the symbol n(B).
The total number of events in a sample space are denoted by n(S).
Different Probability Formulas
When two events, such as A and B, are combined into one, the Probability Formulas says that P(A or B) = P(A) + P(B) – P(ABA∩B).
P(A ∪ B) = P(A B) + P(A∩B)
When an event is the complement of another event, specifically if A is an event, then P(not A) = 1 – P(A) or P(A’) = 1 – P(A) is the complementary Probability Formulas (A).
P(A) + P(A′) = 1.
Probability with the conditional rule: When the Probability Formulas of event B is needed and event A is already known to have occurred, then P(B, given A) = P(A and B), P (A, given B). If event B occurs, it can be the other way around.
P(AB) = P(BA)/P (A)
Probability using a multiplication rule: Whenever two occurrences intersect, or when events A and B must take place at the same time. As a result, P(A and B) equals P(A)P. (B).
P(A) = P(A) P(B) A
Probability Tree Diagram
A tree diagram in Probability is a graphic depiction that aids in determining the likely outcomes or the likelihood that any event will occur or not. It aids in comprehending the outcomes that could result from tossing the coin and determining the likelihood.
Types of Probability
Depending on the outcome or method used to calculate the likelihood that an event will occur, there may be many viewpoints or types of probabilities. There are four different types of probabilities:
In an experiment where there are B equally likely alternatives and event X has exactly A of these outcomes, according to classical probability, also known as the “priori” or “theoretical probability,” the Probability Formulas of X is A/B, or P(X) = A/B. For instance, there are six equally likely possibilities when a fair die is rolled. In other words, there is a 1/6 chance that each number on the die will be rolled.
Through thinking exercises, the experimental or empirical Probability perspective assesses probability. For instance, if a weighted die is rolled and we are unsure which side is heavier, a person can estimate the likelihood of each outcome by counting the number of times they roll the die, determining the percentage of times that outcome occurs, and then calculating the Probability of that outcome.
Subjective probability takes into account a person’s personal expectation that an event will occur. For instance, a fan’s view regarding the likelihood of a specific team winning a football game depends more on their own conviction and emotion than it does on a formal mathematical calculation.
A set of guidelines, or axioms, developed by Kolmogorov are applied to all kinds of axiomatic probability. The following axioms can be used to quantify the likelihood of any event occurring or not.
The chance ranges from zero to one, with one being the highest possibility.
Probability equals one for an occurrence that is guaranteed to occur.
The union of events states that only one of any two mutually exclusive events can occur simultaneously.
Finding the Probability of an Event
The Probability of an occurrence in an experiment is the likelihood that the event will occur. Any event’s probability is a number between (and including) 0 and 1.
Events in Probability
An event in the Probability Formulas Theory is a collection of experiment results or a portion of the sample space.
If an event’s Probability is represented by P(E), then we get
If and only if E is an impossibility, P(E) = 0.
If and only if E is a specific event, then P(E) = 1.
0 ≤ P(E) ≤ 1.
If two occurrences, “A” and “B,” are presented, then event “A” is more likely to occur than event “B” if and only if event “A” is more likely to occur. The number of outcomes in the sample space, n(S), is represented by the set of all potential experiment results, sample space(S).
P(E) = n(E)/n (S)
P(E’) = (1 – (n(E)/n(S))/(n(S) – n(E))
E’ denotes the absence of the occurrence.
As a result, a person may likewise draw the conclusion that P(E) + P(E’) = 1.
Coin Toss Probability
The likelihood of tossing a coin will now be examined. In games like cricket, people frequently toss a coin to choose who will bowl or bat first, and they base their decision on the outcome of the toss. Now it is to be observed how the Probability Formulas notation can be applied to the toss of a single coin. It is also examined how the two and three balls are tossed, respectively.
Tossing a Coin
A single coin flip results in either a head or a tail. The Probability Formulas for receiving the head and getting the tail can be determined using the notion of probability, which is the ratio of positive outcomes to all possible outcomes.
The total number of outcomes is 2, and the sample space is H, T, where H stands for “head” and T for “tail.”
P(H) = Total outcomes / Number of heads = 1/2
P(T) = Total outcomes / Number of Tails = 1/2
Tossing Two Coins
There are a total of four possible results from throwing two coins. The likelihood of two heads, one head, no heads, and a corresponding Probability for the number of tails may all be computed using the Probability Formulas. The probabilities for the two heads are calculated as follows.
Total outcomes are 4, and sample space is (H, H), (H, T), (T, H), and (T, T) respectively.
P(0H) = (2T) = Number of outcomes with two heads/Total Outcomes = 1/4 P(2H) = P(0 T) = Number of outcomes with two heads/Total Outcomes = 1/4 P(1H) = P(1T) = Number of outcomes with only one head/Total Outcomes = 2/4 = 1/2 P(0H) = P(1H) = P(1T) = Number of outcomes with only one head/Total Outcomes
Tossing Three Coins
Tossing three coins at once results in a total of eight outcomes (2 cube). Students can calculate the likelihood of receiving one head, two heads, three heads, or no heads for these outcomes. The quantity of tails can also be determined with a comparable Probability.
Total number of outcomes is 2cube, which equals 8, and Sample Space is (H, H, H), (H, H, T), (T, H, H), (T, T, H), (T, H, T), (H, T, T), (T, T, T),
P(0H) = P(3T) = Number of results with no heads/Total results = 1/8 P(1H) = P(2T) = Number of results with one head/Total results = 3/8 P(2H) = P(1T) = Number of results with two heads/Total results = 3/8 P(3H) = P(0T) = Number of results with three heads/Total results = 1/8
Dice Roll Probability
Dice are frequently used in games to control how players move across the board. The six probable outcomes of a dice are determined by chance and can be calculated using Probability theory. There are several probabilities that may be computed for outcomes when employing two dice, which are also used in some games.
Rolling One Dice
There are six possible results when rolling a dice, and the sample space is 1, 2, 3, 4, 5, and 6. To help with a better grasp of the notion of Probability when rolling a single die, the following few probabilities will be computed.
P(Even Number) equals the number of outcomes with an even number, while
P(Odd Number) equals the number of outcomes with an odd number, and P(Total Outcomes) equals P.
(Prime Number) = Number of prime number outcomes/Total Outcomes
Rolling Two Dice
A square of 6 is equal to 36 total numbers of the possible outcomes when rolling two dice.
Students are required to evaluate a handful of the outcomes from two dice’s probabilities. The following are the Probability.
The odds of getting a doublet with the same number are 6/36, or 1/6.
Probability of at least one die producing the number three is 11/36.
Probability of obtaining 7 is 6/36 (1/6)
As students can see, there are six possible outcomes when they roll a single die. There are 36 possible outcomes when they roll two dice. There are 216 possible outcomes when they roll three dice. Therefore, 6n can be used as a general Probability to describe the number of outcomes from rolling n dice.
Probability of Drawing Cards
The 52 cards in a deck are divided into four suits: spades, clubs, diamonds, and hearts. There are 13 cards in each of the four suits—clubs, diamonds, hearts, and spades—for a total of 52. The likelihood of pulling cards from a pack is: black cards are spades and clubs, red cards are hearts and diamonds.
Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, and king are the 13 cards that make up each suit. The jack, queen, and king are referred to as face cards in these. These examples help students grasp the Probability Formulas.
Drawing a black card has a P(Black card) = 26/52 = 1/2 chance of happening.
Drawing a heart card has a P(Hearts) = 13/52 = 1/4 chance of happening.
Drawing a face card has a chance of P(Face card) = 12/52 = 3/13 chance of happening.
P(4) = 4/52 = 1/13 is the chance of drawing a card with the number 4.
P(4 Red) = 2/52 = 1/26 is the chance of drawing a red card with the number 4.
The following Probability Formulas theorems can be used to do a variety of calculations involving Probability as well as to comprehend the applications of probability.
Theorem 1: The likelihood of an event happening plus the Probability Formulas that it will not happen is equal to one.
P(A) +P(A)= 1
The Probability Formulas of an impossible event or an event not occurring is always equal to 0. This is known as Theorem 2.
The Probability Formulas of a sure event are always equal to 1, according to Theorem 3. P(A) = 1
Theorem 4: Any event’s Probability of occurring always ranges from 0 to 1. 0 P(A) 1
Theorem 5: If there are two events A and B, we may use the union of two sets Probability Formulas to obtain the Probability that either event A or event B will occur, as shown below.
P(A∪B)= P(A)+ P(B)- P(A∩B)
Additionally, we have P(A U B) = P(A) + P for two occurrences A and B that are mutually exclusive P(B).
Bayes’ Theorem on Conditional Probability
The Bayes theorem defines the likelihood of an event based on the assumption that other events will occur. Another name for it is conditional probability. It aids in determining the likelihood that one event will occur given the circumstances under which another event must occur.
Hypothetically, there are three bags, and each one has some blue, green, and yellow balls. How likely is it that a person will choose a yellow ball from the third bag? Since there are also balls that are blue and green, they may calculate the likelihood based on these circumstances as well. Conditional probability is the name given to such a probability.
The Probability Formulas for Bayes’ theorem is P(AIB)= P(BIA). P(A)/ P(B)
P where (AIB)
shows how frequently event A occurs when event B occurs.
P where (BIA)
shows how frequently event B occurs when event A occurs.
P(A) represents the Probability Formulas that event A will occur.
P(B) represents the Probability Formulas that event B will occur.
Law of Total Probability
When there are n events in an experiment, the Probability Formulas of all n events added together is always 1. The sample space can be split into a set of A ∩ B′, A ∩ B, A′ ∩ B, A′ ∩ B′ for two events A and B connected to a sample space S. Any pair of sets in this set is considered pairwise or mutually disjoint because of this. This set’s components are more commonly referred to as a sample space partition.
Solved Examples on Probability
Mathematics provides a way for students to increase their mental discipline and sharpen their logical thinking. Students claim Mathematics is a difficult subject, therefore, students are adviced to study from the solved examples available at the Extramarks’ website. Extramarks not only provide the solved examples but also help students clear their doubts.
Probability Formulas indicate the possibility of the given event. A random event’s occurrence is the subject of this area of Mathematics. The range of the value is 0 to 1. Probability Formulas have been applied into Mathematics to predict the likelihood of different events. Probability generally refers to the degree to which something is likely to occur. This fundamental theorem of Probability Formulas, which also applies to the Probability distribution, can help students comprehend the possible outcomes of a random experiment. Knowing the total number of possible outcomes is necessary before calculating the likelihood that a specific event will occur.
Extramarks is a student-focused digital learning platform. The main goal of the website is to give students the freedom to learn in the way that best suits them.The primary goal is to boost engagement while incorporating graphics to encourage students to study. The Assessment Centre, Smart Class Solutions, and Live Class Platform are a few of the tools that Extramarks provides for use in schools. These tools let teachers include students in learning and design curriculum that is tailored to their requirements. Students can learn more effectively while at home with Extramarks’ live online sessions. Throughout the course, students experiment with various learning tools and platforms, which has accelerated their development of new competencies.
The Extramarks’ website provides solved examples of the Probability Formulas. Students do not need to look elsewhere for the solutions as they can find the step by step Probability Formulas on Extramarks. The examples of the Probability Formulas provided on the website have been chosen by the Extramarks experts to be precise and reliable. The Extramarks provide an explanation of the Probability Formulas examples in depth in order for the students to understand it precisely . For students’ queries and concerns, the Extramarks website provides a variety of tools, such as K12 study materials, live question-answer sessions, and much more. This helps students stay focused on their goals, perform well in their schoolwork, and receive excellent results on their examinations.
The Extramarks organisation is committed to upholding this community of learners’ growth and prosperity. Practice is the first and most crucial phase in preparing for Mathematics since it aids students in organising and developing their core concepts. If students practice the solutions from the solved examples of the Probability Formulas, they will be able to solve any complex problems that come up in their in-class, competitive, or exam situations promptly. Teachers can explain the Probability Formulas examples that are provided on the Extramarks website. It aids students’ grades by emphasising Probability Formulas, in-depth research, and rapid learning.The worksheets and practice questions on the Probability Formulas are available to students from Extramarks in addition to numerous sample papers on the Probability Formulas. Additionally, it keeps tabs on their progress and offers support to the students with queries.
Practice Questions on Probability
Mathematics has used the Probability Formulas to determine the likelihood of certain events. The Probability Formulas, in general, has to do with how likely something is to occur. Students can better understand the potential results of a random experiment by using this fundamental theorem of the Probability Formulas, which also holds true for the probability distribution. Before a student can determine the possibility that a certain event will occur, they must first understand the entire number of alternative possibilities. Most students find the Probability Formulas a complex topic. With the aid of Extramarks’ practice questions with solved answers students can organise their understanding of Probability Formulas and stop battling with Mathematics.
The Probability Formulas have been perceived differently by students, but with Extramarks’ expert guidance, and the practice questions on the Probability Formulas, they will understand the topic. When students use Extramarks and practice, they will be able to achieve the greatest outcomes. For students to learn, practice, and succeed, Extramarks is a website that provides a variety of excellent educational courses, such as solved practice questions. Extramarks’ practice questions on the Probability Formulas, which were created in accordance with the most recent syllabus and examination pattern, offer comprehensive coverage of each and every topic in detail.
The practice questions on the topic Probability Formulas are believed to be the most important for students taking the exam because they cover critical topics. The explanations in the practice questions about the Probability Formulas help students in understanding the concept clearly. The Extramarks’ website provides a range of excellent instructional courses so that students can learn, practice, and perform effectively. When students’ practice and the assistance of Extramarks are combined, students will be able to get the greatest results. Extramarks provides students with practise questions on the Probability Formulas to help them organise their concepts and avoid battling with mathematics.
Students can learn more effectively while at home with Extramarks’ live online sessions. Throughout the course, students experiment with various learning tools and platforms, which has accelerated their development of new competencies. Since students are frequently reluctant to ask questions in class, Extramarks is the safest website for them to get answers about the Probability Formulas. Questions from students are promptly addressed in a way that maximises their learning. Students can learn about their strengths and shortcomings by referring to the Extramarks website, which maintains the whole report. In order to assess students’ progress and help them with their weak areas, teachers also provide them with practice sheets. The whole course syllabus for each subject is available on the Extramarks website.
FAQs (Frequently Asked Questions)
1. What function does probability serve?
Information about the possibility that something will happen is provided by probability. For instance, meteorologists use weather patterns to forecast the likelihood of rain. In statistics, the probability theory is used to understand the relationship between exposures and the risk of health outcomes.
2. How can probability aid in making decisions?
The primary function of probability is to facilitate better decision-making under uncertain conditions. It aids objective, data-driven decision-making as opposed to decision-making based solely on emotion.