Probability Formulas

Probability Formulas

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.

Probability Formula

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:

Classical Probability

Empirical Probability

Subjective Probability

Axiomatic Probability

Classical Probability

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.

Empirical Probability

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

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.

Axiomatic Probability

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.

Probability Theorems

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.

P(ϕ)= 0

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

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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.

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Practice Questions on Probability

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