Probability Formulas

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Probability Formulas

The probability formula is used to compute the probability of an event to occur. To recall, the likelihood of an event happening is called probability. When a random experiment is entertained, one of the first questions that come in our mind is: What is the probability that a certain event occurs? A probability is a chance of prediction. When we assume that, let’s say, x be the chances of happening an event then at the same time (1-x) are the chances for “not happening” of an event.

Similarly, if the probability of an event occurring is “a” and an independent probability is “b”, then the probability of both the event occurring is “ab”. We can use the formula to find the chances of an event happening.

Formula to Calculate Probability

The formula of the probability of an event is:

Probability Formula

Or,

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

Where,

  • P(A) is the probability of an event “A”
  • n(A) is the number of favourable outcomes
  • n(S) is the total number of events in the sample space

Note: Here, the favourable outcome means the outcome of interest.

Sometimes, students get confused about the word “favourable outcome” with “desirable outcome”. In some of the requirements, losing in a certain test or occurrence of an undesirable outcome can be a favourable event for the experiments run.

Basic Probability Formulas

Let A and B are two events. The probability formulas are listed below:

All Probability Formulas List in Maths
Probability Range 0 ≤ P(A) ≤ 1
Rule of Addition P(A∪B) = P(A) + P(B) – P(A∩B)
Rule of Complementary Events P(A’) + P(A) = 1
Disjoint Events P(A∩B) = 0
Independent Events P(A∩B) = P(A) ⋅ P(B)
Conditional Probability P(A | B) = P(A∩B) / P(B)
Bayes Formula P(A | B) = P(B | A) ⋅ P(A) / P(B)

Example Questions Using Probability Formulas

Example 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

Now, by looking at the formula,

Probability of selecting an ace from a deck is,

P(Ace) = (Number of favourable outcomes) / (Total number of favourable outcomes)

P(Ace) = 4/52

= 1/13

So we can say that the probability of getting an ace is 1/13.

Example 2: Calculate the probability of getting an odd number if a dice is rolled.

Solution:

Sample space (S) = {1, 2, 3, 4, 5, 6}

n(S) = 6

Let “E” be the event of getting an odd number, E = {1, 3, 5}

n(E) = 3

So, the Probability of getting an odd number is:

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

= n(E)/n(S)

= 3/6

= ½

Probability

Probability defines the likelihood of occurrence of an event. There are many real-life situations in which we may have to predict the outcome of an event. We may be sure or not sure of the results of an event. In such cases, we say that there is a probability of this event to occur or not occur. Probability generally has great applications in games, in business to make probability-based predictions, and also probability has extensive applications in this new area of artificial intelligence.

The probability of an event can be calculated by probability formula by simply dividing the favorable number of outcomes by the total number of possible outcomes. The value of the probability of an event to happen can lie between 0 and 1 because the favorable number of outcomes can never cross the total number of outcomes. Also, the favorable number of outcomes cannot be negative. Let us discuss the basics of probability in detail in the following sections.

What is Probability?

Probability can be defined as the ratio of the number of favorable outcomes to the total number of outcomes of an event. For an experiment having ‘n’ number of outcomes, the number of favorable outcomes can be denoted by x. The formula to calculate the probability of an event is as follows.

Probability(Event) = Favorable Outcomes/Total Outcomes = x/n

Let us check a simple application of probability to understand it better. Suppose we have to predict about the happening of rain or not. The answer to this question is either “Yes” or “No”. There is a likelihood to rain or not rain. Here we can apply probability. Probability is used to predict the outcomes for the tossing of coins, rolling of dice, or drawing a card from a pack of playing cards.

The probability is classified into theoretical probability and experimental probability.

Terminology of Probability Theory

The following terms in probability help in a better understanding of the concepts of probability.

Experiment: A trial or an operation conducted to produce an outcome is called an experiment.

Sample Space: All the possible outcomes of an experiment together constitute a sample space. For example, the sample space of tossing a coin is head and tail.

Favorable Outcome: An event that has produced the desired result or expected event is called a favorable outcome. For example, when we roll two dice, the possible/favorable outcomes of getting the sum of numbers on the two dice as 4 are (1,3), (2,2), and (3,1).

Trial: A trial denotes doing a random experiment.

Random Experiment: An experiment that has a well-defined set of outcomes is called a random experiment. For example, when we toss a coin, we know that we would get ahead or tail, but we are not sure which one will appear.

Event: The total number of outcomes of a random experiment is called an event.

Equally Likely Events: Events that have the same chances or probability of occurring are called equally likely events. The outcome of one event is independent of the other. For example, when we toss a coin, there are equal chances of getting a head or a tail.

Exhaustive Events: When the set of all outcomes of an experiment is equal to the sample space, we call it an exhaustive event.

Mutually Exclusive Events: Events that cannot happen simultaneously are called mutually exclusive events. For example, the climate can be either hot or cold. We cannot experience the same weather simultaneously.

Probability Formula

The probability formula defines the likelihood of the happening of an event. It is the ratio of favorable outcomes to the total favorable outcomes. The probability formula can be expressed as,

where,

  • P(B) is the probability of an event ‘B’.
  • n(B) is the number of favorable outcomes of an event ‘B’.
  • n(S) is the total number of events occurring in a sample space.

Different Probability Formulas

Probability formula with addition rule: Whenever an event is the union of two other events, say A and B, then
P(A or B) = P(A) + P(B) – P(A∩B)
P(A ∪ B) = P(A) + P(B) – P(A∩B)

Probability formula with the complementary rule: Whenever 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).
P(A) + P(A′) = 1.

Probability formula with the conditional rule: When event A is already known to have occurred and the probability of event B is desired, then P(B, given A) = P(A and B), P(A, given B). It can be vice versa in the case of event B.
P(B∣A) = P(A∩B)/P(A)

Probability formula with multiplication rule: Whenever an event is the intersection of two other events, that is, events A and B need to occur simultaneously. Then P(A and B) = P(A)⋅P(B).
P(A∩B) = P(A)⋅P(B∣A)

Example 1: Find the probability of getting a number less than 5 when a dice is rolled by using the probability formula.

Solution

To find:
Probability of getting a number less than 5
Given: Sample space = {1,2,3,4,5,6}
Getting a number less than 5 = {1,2,3,4}
Therefore, n(S) = 6
n(A) = 4
Using Probability Formula,
P(A) = (n(A))/(n(s))
p(A) = 4/6
m = 2/3

Answer: The probability of getting a number less than 5 is 2/3.

Example 2: What is the probability of getting a sum of 9 when two dice are thrown?

Solution:

There is a total of 36 possibilities when we throw two dice.
To get the desired outcome i.e., 9, we can have the following favorable outcomes.
(4,5),(5,4),(6,3)(3,6). There are 4 favorable outcomes.
Probability of an event P(E) = (Number of favorable outcomes) ÷ (Total outcomes in a sample space)
Probability of getting number 9 = 4 ÷ 36 = 1/9

Answer: Therefore the probability of getting a sum of 9 is 1/9.

Probability Tree Diagram

A tree diagram in probability is a visual representation that helps in finding the possible outcomes or the probability of any event occurring or not occurring. The tree diagram for the toss of a coin given below helps in understanding the possible outcomes when a coin is tossed and thus in finding the probability of getting a head or tail when a coin is tossed.

Types of Probability

There can be different perspectives or types of probabilities based on the nature of the outcome or the approach followed while finding the probability of an event happening. The four types of probabilities are,

  • Classical Probability
  • Empirical Probability
  • Subjective Probability
  • Axiomatic Probability

Classical Probability

Classical probability, often referred to as the “priori” or “theoretical probability”, states that in an experiment where there are B equally likely outcomes, and event X has exactly A of these outcomes, then the probability of X is A/B, or P(X) = A/B. For example, when a fair die is rolled, there are six possible outcomes that are equally likely. That means, there is a 1/6 probability of rolling each number on the die.

Empirical Probability

The empirical probability or the experimental perspective evaluates probability through thought experiments. For example, if a weighted die is rolled, such that we don’t know which side has the weight, then we can get an idea for the probability of each outcome by rolling the die number of times and calculating the proportion of times the die gives that outcome and thus find the probability of that outcome.

Subjective Probability

Subjective probability considers an individual’s own belief of an event occurring. For example, the probability of a particular team winning a football match on a fan’s opinion is more dependent upon their own belief and feeling and not on a formal mathematical calculation.

Axiomatic Probability

In axiomatic probability, a set of rules or axioms by Kolmogorov are applied to all the types. The chances of occurrence or non-occurrence of any event can be quantified by the applications of these axioms, given as,

  • The smallest possible probability is zero, and the largest is one.
  • An event that is certain has a probability equal to one.
  • Any two mutually exclusive events cannot occur simultaneously, while the union of events says only one of them can occur.

Finding the Probability of an Event

In an experiment, the probability of an event is the possibility of that event occurring. The probability of any event is a value between (and including) “0” and “1”.

Events in Probability

In probability theory, an event is a set of outcomes of an experiment or a subset of the sample space.

If P(E) represents the probability of an event E, then, we have,

  • P(E) = 0 if and only if E is an impossible event.
  • P(E) = 1 if and only if E is a certain event.
  • 0 ≤ P(E) ≤ 1.

Suppose, we are given two events, “A” and “B”, then the probability of event A, P(A) > P(B) if and only if event “A” is more likely to occur than the event “B”. Sample space(S) is the set of all of the possible outcomes of an experiment and n(S) represents the number of outcomes in the sample space.

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

P(E’) = (n(S) – n(E))/n(S) = 1 – (n(E)/n(S))

E’ represents that the event will not occur.

Therefore, now we can also conclude that, P(E) + P(E’) = 1

Coin Toss Probability

Let us now look into the probability of tossing a coin. Quite often in games like cricket, for making a decision as to who would bowl or bat first, we sometimes use the tossing of a coin and decide based on the outcome of the toss. Let us check as to how we can use the concept of probability in the tossing of a single coin. Further, we shall also look into the tossing of two and three coming respectively.

Tossing a Coin

A single coin on tossing has two outcomes, a head, and a tail. The concept of probability which is the ratio of favorable outcomes to the total number of outcomes can be used to find the probability of getting the head and the probability of getting a tail.

Total number of possible outcomes = 2; Sample Space = {H, T}; H: Head, T: Tail

  • P(H) = Number of heads/Total outcomes = 1/2
  • P(T)= Number of Tails/ Total outcomes = 1/2

Tossing Two Coins

In the process of tossing two coins, we have a total of four outcomes. The probability formula can be used to find the probability of two heads, one head, no head, and a similar probability can be calculated for the number of tails. The probability calculations for the two heads are as follows.

Total number of outcomes = 4; Sample Space = {(H, H), (H, T), (T, H), (T, T)}

  • P(2H) = P(0 T) = Number of outcome 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) = (2T) = Number of outcome with two heads/Total Outcomes = 1/4

Tossing Three Coins

The number of total outcomes on tossing three coins simultaneously is equal to 23 = 8. For these outcomes, we can find the probability of getting one head, two heads, three heads, and no head. A similar probability can also be calculated for the number of tails.

Total number of outcomes = 23 = 8 Sample Space = {(H, H, H), (H, H, T), (H, T, H), (T, H, H), (T, T, H), (T, H, T), (H, T, T), (T, T, T)}

  • P(0H) = P(3T) = Number of outcomes with no heads/Total Outcomes = 1/8
  • P(1H) = P(2T) = Number of Outcomes with one head/Total Outcomes = 3/8
  • P(2H) = P(1T) = Number of outcomes with two heads /Total Outcomes = 3/8
  • P(3H) = P(0T) = Number of outcomes with three heads/Total Outcomes = 1/8

Dice Roll Probability

Many games use dice to decide the moves of players across the games. A dice has six possible outcomes and the outcomes of a dice is a game of chance and can be obtained by using the concepts of probability. Some games also use two dice, and there are numerous probabilities that can be calculated for outcomes using two dice. Let us now check the outcomes, their probabilities for one dice and two dice respectively.

Rolling One Dice

The total number of outcomes on rolling a die is 6, and the sample space is {1, 2, 3, 4, 5, 6}. Here we shall compute the following few probabilities to help in better understanding the concept of probability on rolling one dice.

  • P(Even Number) = Number of even number outcomes/Total Outcomes = 3/6 = 1/2
  • P(Odd Number) = Number of odd number outcomes/Total Outcomes = 3/6 = 1/2
  • P(Prime Number) = Number of prime number outcomes/Total Outcomes = 3/6 = 1/2

Rolling Two Dice

The total number of outcomes on rolling two dice is 62 = 36. The following image shows the sample space of 36 outcomes on rolling two dice.

Let us check a few probabilities of the outcomes from two dice. The probabilities are as follows.

  • Probability of getting a doublet(Same number) = 6/36 = 1/6
  • Probability of getting a number 3 on at least one dice = 11/36
  • Probability of getting a sum of 7 = 6/36 = 1/6

As we see, when we roll a single die, there are 6 possibilities. When we roll two dice, there are 36 possibilities. When we roll 3 dice we get 216 possibilities. So a general formula to represent the number of outcomes on rolling ‘n’ dice is 6n.

Probability of Drawing Cards

A deck containing 52 cards is grouped into four suits of clubs, diamonds, hearts, and spades. Each of the clubs, diamonds, hearts, and spades have 13 cards each, which sum up to 52. Now let us discuss the probability of drawing cards from a pack. The symbols on the cards are shown below. Spades and clubs are black cards. Hearts and diamonds are red cards.

The 13 cards in each suit are ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king. In these, the jack, the queen, and the king are called face cards. We can understand the card probability from the following examples.

  • The probability of drawing a black card is P(Black card) = 26/52 = 1/2
  • The probability of drawing a hearts card is P(Hearts) = 13/52 = 1/4
  • The probability of drawing a face card is P(Face card) = 12/52 = 3/13
  • The probability of drawing a card numbered 4 is P(4) = 4/52 = 1/13
  • The probability of drawing a red card numbered 4 is P(4 Red) = 2/52 = 1/26

Probability Theorems

The following theorems of probability are helpful to understand the applications of probability and also perform the numerous calculations involving probability.

Theorem 1: The sum of the probability of happening of an event and not happening of an event is equal to 1. P(A)+P(¯A)=1

Theorem 2: The probability of an impossible event or the probability of an event not happening is always equal to 0. P(ϕ)=0

Theorem 3: The probability of a sure event is always equal to 1. P(A) = 1

Theorem 4: The probability of happening of any event always lies between 0 and 1. 0 < P(A) < 1

Theorem 5: If there are two events A and B, we can apply the formula of the union of two sets and we can derive the formula for the probability of happening of event A or event B as follows.

P(A∪B)=P(A)+P(B)−P(A∩B)

Also for two mutually exclusive events A and B, we have P( A U B) = P(A) + P(B)

Bayes’ Theorem on Conditional Probability

Bayes’ theorem describes the probability of an event based on the condition of occurrence of other events. It is also called conditional probability. It helps in calculating the probability of happening of one event based on the condition of happening of another event.

For example, let us assume that there are three bags with each bag containing some blue, green, and yellow balls. What is the probability of picking a yellow ball from the third bag? Since there are blue and green colored balls also, we can arrive at the probability based on these conditions also. Such a probability is called conditional probability.

The formula for Bayes’ theorem is

where, P(A|B)

denotes how often event A happens on a condition that B happens.

where, P(B|A)

denotes how often event B happens on a condition that A happens.

P(A)

the likelihood of occurrence of event A.

P(B)

the likelihood of occurrence of event B.

Law of Total Probability

If there are n number of events in an experiment, then the sum of the probabilities of those n events is always equal to 1.

P(A1)+P(A2)+P(A3)+….P(An)=1

Important Notes on Probability:

Let us check the below points, which help us summarize the key learnings for this topic of probability.

  1. Probability is a measure of how likely an event is to happen.
  2. Probability is represented as a fraction and always lies between 0 and 1.
  3. An event can be defined as a subset of sample space.
  4. The outcome of throwing a coin is a head or a tail and the outcome of throwing dice is 1, 2, 3, 4, 5, or 6.
  5. A random experiment cannot predict the exact outcomes but only some probable outcomes.

Solved Examples on Probability

Example 2: In a bag, there are 6 blue balls and 8 yellow balls. One ball is selected randomly from the bag. Find the probability of getting a blue ball.

Solution:

Let us assume the probability of drawing a blue ball to be P(B)

Number of favorable outcomes to get a blue ball = 6

Total number of balls in the bag = 14

P(B) = Number of favorable outcomes/Total number of outcomes = 6/14 = 3/7

Answer: Therefore the probability of drawing a blue ball is 3/7.

Example 3: There are 5 cards numbered: 2, 3, 4, 5, 6. Find the probability of picking a prime number, and putting it back, you pick a composite number.

Solution:

The two events are independent. Thus we use the product of the probability of the events.

P(getting a prime) = n(favorable events)/ n(sample space) = {2, 3, 5}/{2, 3, 4, 5, 6} = 3/5

p(getting a composite) = n(favorable events)/ n(sample space) = {4, 6}/{2, 3, 4, 5, 6}= 2/5

Thus the total probability of the two independent events= P(prime) × P(composite)

= 3/5 × (2/5)

= 6/25

Answer: Therefore the probability of picking a prime number and a prime number again is 6/25.

Example 4: Find the probability of getting a face card from a standard deck of cards using the probability formula.

Solution: To find:
Probability of getting a face card
Given: Total number of cards = 52
Number of face cards = Favorable outcomes = 12
Using Probability Formula,
Probability = (Favorable Outcomes)÷(Total Favourable Outcomes)
P(face card) = 12/52
m = 3/13

Answer: The probability of getting a face card is 3/13

FAQs on Probability

What is Probability?

Probability is a branch of math which deals with finding out the likelihood of the occurrence of an event. Probability measures the chance of an event happening and is equal to the number of favorable events divided by the total number of events. The value of probability ranges between 0 and 1, where 0 denotes uncertainty and 1 denotes certainty.

How To Calculate Probability Using the Probability Formula?

The probability of any event depends upon the number of favorable outcomes and the total outcomes. In general, the probability is the ratio of the number of favorable outcomes to the total outcomes in that sample space. It is expressed as, Probability of an event P(E) = (Number of favorable outcomes) ÷ (Sample space).

How to Determine Probability?

The probability can be determined by first knowing the sample space of outcomes of an experiment. A probability is generally calculated for an event (x) within the sample space. The probability of an event happening is obtained by dividing the number of outcomes of an event by the total number of possible outcomes or sample space.

What are the Three Types of Probability?

The three types of probabilities are theoretical probability, experimental probability, and axiomatic probability. The theoretical probability calculates the probability based on formulas and input values. The experimental probability gives a realistic value and is based on the experimental values for calculation. Quite often the theoretical and experimental probability differ in their results. And the axiomatic probability is based on the axioms which govern the concepts of probability.

What is Conditional Probability?

The conditional probability predicts the happening of one event based on the happening of another event. If there are two events A and B, conditional probability is a chance of occurrence of event B provided the event A has already occurred. The formula for the conditional probability of happening of event B, given that event A, has happened is P(B/A) = P(A ∩ B)/P(A).

What is Experimental Probability?

The experimental probability is based on the results and the values obtained from the probability experiments. Experimental probability is defined as the ratio of the total number of times an event has occurred to the total number of trials conducted. The results of the experimental probability are based on real-life instances and may differ in values from theoretical probability.

What is a Probability Distribution?

The two important probability distributions are binomial distribution and Poisson distribution. The binomial distribution is defined for events with two probability outcomes and for events with a multiple number of times of such events. The Poisson distribution is based on the numerous probability outcomes in a limited space of time, distance, sample space. An example of the binomial distribution is the tossing of a coin with two outcomes, and for conducting such a tossing experiment with n number of coins. A Poisson distribution is for events such as antigen detection in a plasma sample, where the probabilities are numerous.

How are Probability and Statistics Related?

The probability calculates the happening of an experiment and it calculates the happening of a particular event with respect to the entire set of events. For simple events of a few numbers of events, it is easy to calculate the probability. But for calculating probabilities involving numerous events and to manage huge data relating to those events we need the help of statistics. Statistics helps in rightly analyzing

How Probability is Used in Real Life?

Probability has huge applications in games and analysis. Also in real life and industry areas where it is about prediction we make use of probability. The prediction of the price of a stock, or the performance of a team in cricket requires the use of probability concepts. Further, the new technology field of artificial intelligence is extensively based on probability.

How was Probability Discovered?

The use of the word probable started first in the seventeenth century when it was referred to actions or opinions which were held by sensible people. Further, the word probable in the legal content was referred to a proposition that had tangible proof. The field of permutations and combinations, statistical inference, cryptoanalysis, frequency analysis have altogether contributed to this current field of probability.

Where Do We Use the Probability Formula In Our Real Life?

The following activities in our real-life tend to follow the probability formula:

  • Weather forecasting
  • Playing cards
  • Voting strategy in politics
  • Rolling a dice.
  • Pulling out the exact matching socks of the same color
  • Chances of winning or losing in any sports.

What is the Conditional Probability Formula?

The conditional probability depends upon the happening of one event based on the happening of another event. The conditional probability formula of happening of event B, given that event A, has already happened is expressed as P(B/A) = P(A ∩ B)/P(A).

Formula to Calculate Probability

You must have heard the term “probability” being coined for predicting the weather forecast in news TV bulletins for the next few days for some parts of the country. For calculating the probability of different types of situations, the probability formula and its related basic concepts are used. Probability is the way to measure the uncertainty, how likely an event has happened or is bound to happen.

Probability is that branch of mathematics that is concerned with the numerical description of how likely there are chances of the event to occur or how likely a particular proposition is true. For any event the probability lies between 0 to 1. 0 indicates the impossibility of the event to happen while 1 indicates certainty that the event is certain to occur.

Notation of Probability

For example: let us consider that two events are taking place namely A and B. So for the probability that event A can happen, we are going to write P(A) and for the probability that event B can happen, we can write P(B).

Terminologies Related to Probability Formula

There are a few crucial terminologies that are associated with all probability formulas

  • Experiment: Any particular situation or an event for which we are required to find the probability is known as an experiment. Rolling a dice, tossing a coin are the most simple examples we can use.
  • Outcome: The result of an event after experimenting with the side of the coin after flipping, the number appearing on dice after rolling and a card is drawn out from a pack of well-shuffled cards, etc.
  • Event: The combination of all possible outcomes of an experiment like getting head or tail on a tossed coin, getting an even or odd number on dice, etc.
  • Sample Space: The set of all possible results or outcomes. This indicates that besides this there is no chance that any other result will come. Whatever the result is, it is from this sample Space only.
  • Probability Function: The function helps in obtaining the probability of every outcome.

Probability Formulas with Examples

To Calculate the probability of an event to occur we use this probability formula, recalling, the probability is the likelihood of an event to happen. This formula is going to help you to get the probability of any particular event. 

This formula is the number of favourable outcomes to the total number of all the possible outcomes that we have already decided in the Sample Space.

The probability of an Event = (Number of favourable outcomes) / (Total number of possible outcomes)

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

P(A) < 1

Here, P(A) means finding the probability of an event A, n(E) means the number of favourable outcomes of an event and n(S) means the set of all possible outcomes of an event.

If the probability of occurring an event is P(A) then the probability of not occurring an event is

P(A’) = 1- P(A)

Example 01: Probability of obtaining an odd number on rolling dice for once.

Solution: Sample Space = {1, 2, 3, 4, 5, 6}

n(S) = 6

Favourable outcomes = {1, 3, 5}

n(E) = 3

Using the probability formula,

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

P(Getting an odd number) = 3 / 6 = ½ = 0.5

Important List of Probability Formulas

You just need to have the events for which you are looking for the probability and the formulas are going to make your work easier. In the formulas given below, we are taking 2 events namely A and B. The formulas are based on these events only.

  • Event (A OR B)

P (A U B) = P (A) + P (B) – P (A ∩ B)

  • Event (A AND B)

P (A ∩ B) = P (A) . P (B)

  • Event (A NOT B)

P(A NOT B) = A – B 

  • Event (B NOT A)

P(B NOT A) = B – A

  • Event (NOT A)

Probability of occurrence of an event is P(A)

Probability of non-occurrence of the same event is P(A’). 

Some probability important formulas based on them are as follows:

  • 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(AB’) + P(A’B) + P(A.B)

Example 01: Two dice are rolled simultaneously. Calculate the probability of getting the sum of the numbers on the two dice is 6.

Solution: 

Sample Space= (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6){(1, 1), (1, 2), (1,3), (1,4), (1,5), (1, 6)} {(2, 1), (2, 2),(2,3), (2,4), (2,5), (2, 6)} {(3, 1), (3, 2), (3,3), (3,4), (3,5), (3, 6)} {(4, 1), (4, 2), (4,3), (4,4), (4,5), (4, 6)} {(5, 1), (5,2), (5,3), (5,4), (5,5), (5, 6)} {(6, 1), (6, 2), (6,3), (6,4), (6,5), (6, 6)} n(S) = 36 

Favourable outcomes = {(1, 5), (2, 4), (3, 3), (4, 2) and (5, 1)} 

n(E) = 5

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

P(Getting sum of numbers on two dice 6) = 5/ 36

FAQs (Frequently Asked Questions)

1. Is probability a difficult topic in maths?

No. Probability is one of the most interesting topics covered in school level mathematics. It has several applications in the advanced concepts of mathematics and statistics. There are a few formulas that students need to learn and practice to develop a good understanding of the concepts and applications of Probability. This concept seems to be a little different from the rest of the topics covered in the syllabus for Class 10 and 12 maths, but with good practice, students can easily master its applications. The topic of Probability carries a considerable weightage in both Class 10 and Class 12 Mathematics examinations.

2. Where can I find good study resources for the topic of probability?

You can find very good study resources for the topic of Probability on Vedantu, for both 10th and 12th-grade syllabi. There are NCERT solutions from the topic probability available on our website and mobile application. These solutions are prepared by the subject matter experts at Vedantu, in strict adherence to the CBSE guidelines. Hence you can refer to the stepwise solutions for a better understanding of the concept of Probability.

3. Can I download the important notes on probability formula for free?

Yes, you can download the important notes for the Probability Formula free of cost from Vedantu. These notes are available on our webpage and you can also download the NCERT solutions for Probability from our mobile application or website for free. These solutions will help you with a deeper knowledge of the basic concepts of Probability, thereby, making it a hassle-free learning experience for all students.

4. What are the common formulas used in probability sums?

You will commonly find sums on rolling of dice, tossing a pair of coins, pack of cards, drawing of different coloured balls or marbles from a bag, etc, from the topic of Probability. It is an added advantage if you have a good concept of set theory, to understand the sums of Probability. Some of the formulas that are commonly used in these sums are as follows.

  • P(A’)= 1- P(A)
  • P(A)= n(E)/n(S)
  • 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(AB’) + P(A’B) +P(A.B)

It is very important to have a clear understanding of the mathematical applications of these above formulas, to solve the sums of this topic. There are applications of permutation and combinations in some sums of Probability, as well.

5. What is the easiest way by which students can understand probability?

To understand the probability concepts easily, first, the students need to go through the solved question papers and the examples of probability. After having a look at the solved papers and examples, students should go with understanding the basics of probability. Then they should look out for the formulas and other examples that Vedantu provides you side by side so that you are well aware of the application of the concept that you have studied. For helping the students many mock tests are available at Vedantu that will help the students to get a better knowledge of this topic.

Solved Examples on Probability Formulas Class 12

Here we have provided some probability solved examples.

Question 1: A coin is thrown 3 times. What is the probability that at least one head is obtained?

Solution: Sample space = [HHH, HHT, HTH, THH, TTH, THT, HTT, TTT]
Total number of ways = 2 × 2 × 2 = 8.
Favourable Cases = 7 [Need at least 1 Head]
P (A) = No. of Favourable Outcomes/Total number of outcomes
= 7/8

Question 2: Two cards are drawn from the pack of 52 cards. Find the probability that both are diamonds or both are kings.

Solution: Total no. of ways = 52C2
Case I: Both are diamonds = 13C2
Case II: Both are kings = 4C2 
P (both are diamonds or both are kings) = (13C2 + 4C2 ) / 52C2 = 14/221

Question 3: Calculate the probability of getting an even number if a dice is rolled.

Solution: Sample space (S) = {1, 2, 3, 4, 5, 6}
n(S) = 6
Let “E” be the event of getting an odd number, E = {2, 4, 6}
n(E) = 3
So, the Probability of getting an odd number is:
P(E) = (Number of outcomes favorable)/(Total number of outcomes)
= n(E)/n(S)
= 3/6
= 1/2

Question 4: What is the probability of getting a sum of 22 or more when four dice are thrown?

Solution: Total number of ways = 64 
= 1296
(i) Number of ways of getting a sum 22 are 6,6,6,4 = 4! / 3!
= 4 and 6,6,5,5 = 4! / 2!2!
= 6.
(ii) Number of ways of getting a sum 23 is 6,6,6,5 = 4! / 3! = 4
(iii) Number of ways of getting a sum 24 is 6,6,6,6 = 1.
Fav. Number of cases = 4 + 6 + 4 + 1 = 15 ways.
P (getting a sum of 22 or more) = 15/1296
= 5/432

Question 5: Find the probability that a leap year has 52 Sundays.

Solution: A leap year can have 52 Sundays or 53 Sundays.
In a leap year, there are 366 days out of which there are 52 complete weeks & the remaining 2 days.
Now, these two days can be (Sat, Sun), (Sun, Mon), (Mon, Tue), (Tue, Wed), (Wed, Thu), (Thu, Fri), (Fri, Sat).
So there are a total of 7 cases out of which (Sat, Sun) (Sun, Mon) are two favourable cases.
So,  P (53 Sundays) = 2 / 7
Now, P(52 Sundays) + P(53 Sundays) = 1
So, P (52 Sundays) = 1 – P(53 Sundays) = 1 – (2/7) = (5/7)

Practice Questions on All Probability Formulas

Here we have provided some of the practice questions for probability formulas for Class 7 for you to practice:

Question 1: Three bags contain 3 red, 7 black; 8 red, 2 black, and 4 red & 6 black balls respectively. 1 of the bags is selected at random and a ball is drawn from it. If the ball drawn is red, find the probability that it is drawn from the third bag.

Question 2: Fifteen people sit around a circular table. What are the odds against two particular people sitting together?

Question 3: From a pack of cards, three cards are drawn at random. Find the probability that each card is from a different suit.

Question 4: Two dice are thrown together. What is the probability that the number obtained on one of the dice is multiple of the number obtained on the other dice?

Question 5: 1 card is drawn at random from the pack of 52 cards.
(i) Find the probability that it is an honour card.
(ii) It is a face card.

Question 6: There are 5 green and 7 red balls. Two balls are selected one by one without replacement. Find the probability that the first is green and the second is red.

Question 7: Consider another example where a pack contains 4 blue, 2 red and 3 black pens. If a pen is drawn at random from the pack, replaced and the process repeated 2 more times, What is the probability of drawing 2 blue pens and 1 black pen?

Question 8: In a class, 40% of the students study math and science. 60% of the students study math. What is the probability of a student studying science given he/she is already studying math?

Frequently Asked Questions

Here we have provided some of the frequently asked questions related to statistical probability formulas:

Q1: Where can I download probability formulas for aptitude?
A: You can download the probability formulas for aptitude PDF from Embibe. We have provided all the basic formulas along with probability equations on this page.

Q2: Which platform is best for beginners to learn probability basics?
A: Embibe is one of the best education technologies that offer basic to next-level probability tutorials for students. Students can also take get access to the list of all probability formulae on Embibe and try some mock tests to be ready for their exams. 

Q3: What is the easiest way to understand probability?
A: The easiest way to understand probability is to first take a look at the solved question papers and the probability examples. After that, students should start with the basics of probability. There are many mock tests available on Embibe for the probability that students can attempt to clear their doubts about probability. 

Q4: What is the formula of probability?
A: The probability of an event is the number of favourable outcomes divided by the total number of outcomes possible. This basic definition of probability assumes that all the outcomes are equally likely to occur.

Q5: What are the 3 types of probability?
A: There are 3 types of probability:
(i) Theoretical Probability.
(ii) Experimental Probability.
(iii) Axiomatic Probability.

Q6: What does P (AUB) mean?
A: P(A U B) is the probability of the sum of all sample points in A U B. It is defined by P(A) + P(B), which is the sum of probabilities of sample points in A and in B.

Q7: What does P (A|B) mean?
A: P(A|B) is the conditional probability. It is the probability of event A occurring, given that event B occurs.

Now you are provided with all the necessary information regarding all probability formulas in Class 9. We hope that you have downloaded the PDF of probability formulas available on this page. Practice more questions and master this concept.

Example 1: Probability of getting an even number on rolling a dice once.
Solution: Sample Space (S) = {1, 2, 3, 4, 5, 6}
Event (E) = {2, 4, 6}
Therefore, n (S) = 6 and n (E) = 3
Putting this in the probability formula, we get:
P = 3 / 6 = 1 / 2 = 0.5
This means, that the chances of getting an even number upon rolling a dice is 0.5

Example 2: Probability of getting HEAD at least once on tossing a coin twice.
Solution: Sample Space (S) = {HH, HT, TH, TT}; where H denotes Head and T denotes Tail.
Event (E) = {HH, HT, TH}
Therefore, Therefore, n (S) = 4 and n (E) = 3
Putting this in the probability formula, we get:
P = 3 / 4 = 0.75
This means, that the chances of getting at least one HEAD on tossing a coin twice are 0.75

Odds in Favour of the Event

Odds in the favor of any event is the ratio of the number of ways that an outcome can occur to the number of ways it cannot occur. Let’s look at an example.

Example 3: If a represents the odds in favor of getting number 4 on a single roll of dice & b represents the outcomes of not getting 4, then,

n (a) = Number of favorable outcomes = 1
n (b) = Number of favorable outcomes = (6 – 1) = 5
Odds in favor = 1 : 5 or 1 / 5
Probability (P) = Number of favorable outcomes/(Number of favorable outcomes + Number of unfavorable outcomes)
P = 1 / (1 + 5) = 1 / 6

Odds Against the Event

Odds against any event is the ratio of the number of ways that an outcome cannot occur to the number of ways it can occur. Let’s understand it through an example.

Example 4:  If a represents the odds against getting number 4 on a single roll of dice & b represents the outcomes of getting 4, then –

n (a) = Number of favorable outcomes = 1
n (b) = Number of favorable outcomes = (6 – 1) = 5
Odds in favor = 5 : 1 or 5 / 1
Probability (P) = Number of favorable outcomes / (Number of favorable outcomes + Number of unfavorable outcomes)
P = 5 / (1 + 5) = 5 / 6

Important Probability Formulas

I. Event (A OR B)

Also given by P (A U B) = P (A) + P (B) – P (A ∩ B)

If A & B are two mutually exclusive events then P (A ∩ B) = 0 and P (A U B) = P (A) + P (B). For example,

A = {Numbers greater than or equal to 4 in a dice roll} = {4, 5, 6}
B = {Numbers lesser than or equal to 4 in a dice roll} = {1, 2, 3, 4}
Thus, (A U B) = P (A) + P (B) = {1, 2, 3, 4, 5, 6}

II. Event (A AND B)

Also given by P (A ∩ B) = P (A) . P (B)

It gives the common elements that form the individual subsets of events A and B. For example,

A = {Numbers greater than or equal to 4 in a dice roll} = {4, 5, 6}
B = {Numbers lesser than or equal to 4 in a dice roll} = {1, 2, 3, 4}
Thus, (A ∩ B) = P (A) . P (B) = {4}

III. Event (A but NOT B)

For example,
A = {Numbers greater than or equal to 4 in a dice roll} = {4, 5, 6}
B = {Numbers lesser than or equal to 4 in a dice roll} = {1, 2, 3, 4}
Thus, (A but NOT B) = A – B = {1, 2, 3}; elements common in A and B get eliminated from A.

IV. Event (B but NOT A)

For example,
A = {Numbers greater than or equal to 4 in a dice roll} = {4, 5, 6}
B = {Numbers lesser than or equal to 4 in a dice roll} = {1, 2, 3, 4}
Thus, (B but NOT A) = B – A = {5, 6}; elements common in A and B get eliminated from B.

V. Event (NOT A)

The probability of occurrence of an event – P(A) then the probability of non-occurrence of the same event is P(A’). Some probability formulas based on them are as follows:

  • 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(AB’) + P(A’B) + P(A.B)

VI. Conditional Probability

P (B/A):Probability (conditional) of event B when event A has occurred.
P (A/B):Probability (conditional) of event A when event B has occurred.
P (A ∩ B) = P (A) . P (B/A)

These are some of the formulas that will help you solve mathematical problems on Probability.

Solved examples for You

Question: Find the probability of getting an even number greater than or equal to 4 in a dice roll.

Solution: Sample space (S) = {1, 2, 3, 4, 5, 6} and E = {4, 6}
P (E) = n (E) / n (S)
= 2 / 6
P (E) = 1 / 3

Question: Find the probability of getting at least one HEAD in a double coin toss.

Solution: S = {HH, HT, TH, TT}
E = {HH, TH, HT}
P (E) = n (E) / n(S)
So, P (E) = 3 / 4

This concludes our discussion on the topic of probability formulas.

Question: What is a simple probability?

Answer: It refers to the ratio of a number of outcomes favourable for an event to the total number of possible outcomes. In simple words, it is the measure of the possibility of an event or measure of the chance of an event to happen.

Question: What do U and ∩ mean in probability?

Answer: U simply means uniform distribution in probability. On the other hand, ∩ refers to the intersection of sets: Besides, the intersection of two given sets is the largest set that covers all the elements that are common to both the sets.

Question: State the four rules of probability?

Answer: Four rules of probability are:

  1. Happening or non-happening: the possibility of happening and non-happening is always 1.
  2. Exclusivity: Means that if two events A and B happen at the same time then they are mutually exclusive
  3. Independence: Means that when one happens the other will not happen and visa-versa.
  4. Sub-events: When one happens the other also happen.

Question: State the types of probability distributions?

Answer: Some of the types of probability distributions are:

  1. Normal distribution
  2. Chi-Square distribution
  3. Binomial distribution
  4. Poisson distribution

However, each probability distribution serves a different purpose and represent different data generation process.

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