Probability Formulas

Probability defines the likelihood of the occurrence of an event. Probability formulas are essential mathematical tools used for calculating the likelihood of events. Before covering its formulas, it’s important to grasp the concept of probability. Probability quantifies the likelihood of a random event occurring. Essentially, it represents the chance of a particular outcome. Probability has wide-ranging applications, from gaming strategies and business forecasting to advancements in artificial intelligence.

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 Formula

Probability formulas determine the likelihood of an event by dividing the number of favorable outcomes by the total number of possible outcomes. This approach allows us to estimate the probability of a specific occurrence.

Mathematically, the formula is expressed as:

P(E) = Number of favorable outcomes/Total number of possible outcomes

The probability formula calculates the ratio of favorable outcomes to the total set of possible outcomes. The resulting probability value ranges from 0 to 1, indicating that the number of favorable outcomes cannot exceed the total number of outcomes, and negative values for favorable outcomes are not possible.

Terms Related to Probability Formula

Here are some common terms related to the probability formula:

Experiment: An action or procedure conducted to generate a specific outcome.

Sample Space: The set of all possible outcomes from an experiment. For instance, the sample space for flipping a coin includes {head, tail}.

Favorable Outcome: The result that aligns with the intended or expected conclusion. For example, favorable outcomes for rolling two dice to get a sum of 4 are (1,3), (2,2), and (3,1).

Trial: The execution of a random experiment.

Random Experiment: An experiment with a well-defined set of possible outcomes, such as tossing a coin, where the result could be either heads or tails, making the outcome uncertain.

Event: The total outcomes resulting from a random experiment.

Equally Likely Events: Events that have identical probabilities of occurring. The outcome of one event does not affect the outcome of another.

Exhaustive Events: Events where the set of all possible outcomes covers the entire sample space.

Mutually Exclusive Events: Events that cannot occur simultaneously. For example, when tossing a coin, the result will be either heads or tails, but not both at the same time.

In probability theory, an event is a set of possible outcomes resulting from an experiment and is often a subset of the total sample space. Denoting the probability of an event E as P(E), the following principles apply:

Events in Probability Formula

• If event E is impossible, P(E)=0.
• If event E is certain, P(E)=1.

The probability P(E) always falls between 0 and 1.

Different Probability Formulas

The different Probability Formulas are given below:

Classical Probability Formula

P(A) = Number of Favorable Outcomes/Total Number of Possible Outcomes

Addition Rule Formula

When considering an event that is the union of two distinct events, A and B, the probability of this union is given by:

P(A or B) = P(A)+P(B)−P(A∩B)

Alternatively, this can be expressed as:

P(A∪B)=P(A)+P(B)−P(A∩B)

Addition Rule for Mutually Exclusive Events

If events A and B are mutually exclusive, meaning they cannot occur simultaneously, the probability of either event occurring is the sum of their individual probabilities:

P(A or B) = P(A)+P(B)

Complementary Rule Formula

If A is an event, the probability of not being A is expressed by the complementary rule:

P(not A) = 1−P(A)

or

P(A′) = 1 − P(A)

Thus,

P(A)+P(A′)=1

Some probability formulas based on this are:

P(A⋅A′)=0

P(A⋅B)+P(A′⋅B′)=1

P(A′⋅B)=P(B)−P(A⋅B)

P(A⋅B′)=P(A)−P(A⋅B)

P(A+B)=P(A⋅B′)+P(A′⋅B)+P(A⋅B)

Joint Probability Formula

It represents the common elements shared by both events A and B. The formula for this can be expressed as:

P(A∩B)=P(A)⋅P(B)

Conditional Rule Formula

When the occurrence of event A is already known, the probability of event B occurring is referred to as conditional probability. It can be calculated using the formula:

P(B∣A)= P(A∩B)/P(A)

​Where:

• P(B∣A) is the conditional probability of event B given that event A has occurred.
• P(A∣B) is the conditional probability of event A given that event B has occurred.

Probability Formula with the Multiplication Rule

In situations where an event represents the simultaneous occurrence of two other events, A and B, the probability of both events happening together can be calculated using the following formulas:

P(A ∩ B) = P(A)⋅P(B) (in case of independent events)

P(A∩B) = P(A)⋅P(B∣A) (in case of dependent events)

Disjoint Event

Disjoint events, also referred to as mutually exclusive events, are events that never occur simultaneously.

This can be expressed as: P(A∩B)=0

Bayes’ Theorem

Bayes’ Theorem calculates the probability of event A occurring given the prior occurrence of event B. This theorem is expressed through the following formula:

P(A∣B)= P(B∣A)×P(A)/ P(B)

Dependent Probability Formula

Dependent probability refers to events whose outcomes are influenced by the occurrence of other events. This concept is encapsulated in the formula for dependent probability, which expresses the likelihood of one event given the presence or outcome of another event.

P(B and A) = P(A)×P(B | A)

Independent Probability Formula

Independent probability pertains to events whose outcomes are not influenced by the occurrence of other events. This principle is captured by the formula for independent probability, which calculates the likelihood of one event occurring without regard to the outcomes of other events.

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 “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.

How to Calculate Probability

The probability of an event can be determined using the formula:

Probability of an Event= Count of favorable outcomes/’Total number of possible outcomes for the event

This is expressed mathematically as:

P(A)=n(E)/n(S)

where 

• P(A) represents the probability of event A,
• n(E) is the count of favorable outcomes,
• n(S) is the total number of possible outcomes for the event.

The probability value, P(A), will always be less than 1.

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) = Number of even outcomes/Total number of possible outcomes = 3/6 = 1/2
• P(Odd Number) = Number of odd outcomes/Total number of possible outcomes = 3/6 = 1/2

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 probabilities:.

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.

• P(Black card) = 26/52 = 1/2 chance of happening.
• P(Hearts) = 13/52 = 1/4 chance of happening.
• P(Face card) = 12/52 = 3/13 chance of happening.
• P(getting 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

Theorem 2: The Probability Formulas of an impossible event or an event not occurring is always equal to 0. This is known as 

P(ϕ)= 0

Theorem 3: The Probability Formulas of a sure event are always equal to 1 i.e. 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 of 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.

 Probability Formulas for Bayes’ theorem is P(A|B) = P(B|A). P(A)/ P(B) 

where

• P(A|B) shows how frequently event A occurs when event B occurs.
• P(B|A) 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. 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.

P(E1) + P(E2) + P(E3) + … + P(En) = 1

Solved Examples on Probability

Example 1: In a bag containing a total of 12 bulbs, consisting of 2 green bulbs, 4 orange bulbs, and 6 white bulbs, if a bulb is selected randomly, what is the probability of picking either a green bulb or a white bulb?

Solution: 

The total number of bulbs in the bag is 12, comprising 2 green bulbs and 6 white bulbs. To find the probability of picking either a green bulb or a white bulb, we add the number of green bulbs to the number of white bulbs and then divide by the total number of bulbs:

Probability = (Number of green bulbs + Number of white bulbs) / Total number of bulbs

Substituting the given values:

Probability = (2 + 6) / 12

Probability = 8 / 12

Probability = 2 / 3

Example 2: When two dice are thrown, what is the probability of obtaining a sum of 9?

Solution: 

When two dice are thrown, there are a total of 36 possible outcomes. To achieve a sum of 9, the favorable outcomes are (4,5), (5,4), (6,3), and (3,6), totaling 4 favorable outcomes. Therefore, the probability of obtaining a sum of 9 is calculated as follows:

P(E) = number of favorable outcomes/Total outcomes in the sample space

​Substituting the values:

P(E)=4/36

​Simplifying:

P(E)=1/9

Example 3: ​If P(E) = 0.25, What is the probability of ‘not E’?

Solution: 

We already know that

P(E) + P(not E) = 1

P(E) = 0.25

So, P(not E) = 1 – P(E)

P(not E) = 1 – 0.25

Hence, P (not E) = 0.75

Practice Questions on Probability

Q1. From a bag containing marbles—8 red, 9 blue, and 6 green—two marbles are randomly selected without replacement. What is the probability that both marbles chosen are blue?

Q2. In a drawer containing pens—6 black, 4 blue, and 7 red—a pen is drawn at random. What is the probability that the pen drawn is either black or blue?

Q3. Find the likelihood of selecting a good ball from a group of balls where 10 are defective and 144 are good. The selection is made randomly, without prior knowledge of whether a ball is defective or not.

Q4. Determine the probability that a leap year will have 52 Sundays.