maths assignment on probability

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Probability

Probability  means possibility. It is a branch of mathematics that deals with the occurrence of a random event. The value is expressed from zero to one. Probability has been introduced in Maths to predict how likely events are to happen. The meaning of probability is basically the extent to which something is likely to happen. This is the basic probability theory, which is also used in the  probability distribution , where you will learn the possibility of outcomes for a random experiment. To find the probability of a single event to occur, first, we should know the total number of possible outcomes.

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Probability Definition in Math

Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur i.e., how likely they are going to happen, using it. Probability can range from 0 to 1, where 0 means the event to be an impossible one and 1 indicates a certain event. Probability for Class 10 is an important topic for the students which explains all the basic concepts of this topic. The probability of all the events in a sample space adds up to 1.

For example , when we toss a coin, either we get Head OR Tail, only two possible outcomes are possible (H, T). But when two coins are tossed then there will be four possible outcomes,  i.e {(H, H), (H, T), (T, H),  (T, T)}.

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Formula for Probability

The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favourable outcomes and the total number of outcomes.

Sometimes students get mistaken for “favourable outcome” with “desirable outcome”. This is the basic formula. But there are some more formulas for different situations or events.

Solved Examples

1) There are 6 pillows in a bed, 3 are red, 2 are yellow and 1 is blue. What is the probability of picking a yellow pillow?

Ans: The probability is equal to the number of yellow pillows in the bed divided by the total number of pillows, i.e. 2/6 = 1/3.

2) There is a container full of coloured bottles, red, blue, green and orange. Some of the bottles are picked out and displaced. Sumit did this 1000 times and got the following results:

• No. of blue bottles picked out: 300
• No. of red bottles: 200
• No. of green bottles: 450
• No. of orange bottles: 50

a) What is the probability that Sumit will pick a green bottle?

Ans: For every 1000 bottles picked out, 450 are green.

Therefore, P(green) = 450/1000 = 0.45

b) If there are 100 bottles in the container, how many of them are likely to be green?

Ans: The experiment implies that 450 out of 1000 bottles are green.

Therefore, out of 100 bottles, 45 are green.

Probability Tree

The tree diagram helps to organize and visualize the different possible outcomes. Branches and ends of the tree are two main positions. Probability of each branch is written on the branch, whereas the ends are containing the final outcome. Tree diagrams are used to figure out when to multiply and when to add. You can see below a tree diagram for the coin:

Types of Probability

There are three major types of probabilities:

Theoretical Probability

Experimental probability, axiomatic probability.

It is based on the possible chances of something to happen. The theoretical probability is mainly based on the reasoning behind probability. For example, if a coin is tossed, the theoretical probability of getting a head will be ½.

It is based on the basis of the observations of an experiment. The experimental probability can be calculated based on the number of possible outcomes by the total number of trials. For example, if a coin is tossed 10 times and head is recorded 6 times then, the experimental probability for heads is 6/10 or, 3/5.

In axiomatic probability, a set of rules or axioms are set which applies to all types. These axioms are set by Kolmogorov and are known as Kolmogorov’s three axioms. With the axiomatic approach to probability, the chances of occurrence or non-occurrence of the events can be quantified. The axiomatic probability lesson covers this concept in detail with Kolmogorov’s three rules (axioms) along with various examples.

Conditional Probability is the likelihood of an event or outcome occurring based on the occurrence of a previous event or outcome.

Probability of an Event

Assume an event E can occur in r ways out of a sum of n probable or possible equally likely ways . Then the probability of happening of the event or its success is expressed as;

The probability that the event will not occur or known as its failure is expressed as:

P(E’) = (n-r)/n = 1-(r/n)

E’ represents that the event will not occur.

Therefore, now we can say;

P(E) + P(E’) = 1

This means that the total of all the probabilities in any random test or experiment is equal to 1.

What are Equally Likely Events?

When the events have the same theoretical probability of happening, then they are called equally likely events. The results of a sample space are called equally likely if all of them have the same probability of occurring. For example, if you throw a die, then the probability of getting 1 is 1/6. Similarly, the probability of getting all the numbers from 2,3,4,5 and 6, one at a time is 1/6. Hence, the following are some examples of equally likely events when throwing a die:

• Getting 3 and 5 on throwing a die
• Getting an even number and an odd number on a die
• Getting 1, 2 or 3 on rolling a die

are equally likely events, since the probabilities of each event are equal.

Complementary Events

The possibility that there will be only two outcomes which states that an event will occur or not. Like a person will come or not come to your house, getting a job or not getting a job, etc. are examples of complementary events. Basically, the complement of an event occurring in the exact opposite that the probability of it is not occurring. Some more examples are:

• It will rain or not rain today
• The student will pass the exam or not pass.
• You win the lottery or you don’t.

Also, read: 

• Independent Events
• Mutually Exclusive Events

Probability Theory

Probability theory had its root in the 16th century when J. Cardan, an Italian mathematician and physician, addressed the first work on the topic, The Book on Games of Chance. After its inception, the knowledge of probability has brought to the attention of great mathematicians. Thus, Probability theory is the branch of mathematics that deals with the possibility of the happening of events. Although there are many distinct probability interpretations, probability theory interprets the concept precisely by expressing it through a set of axioms or hypotheses. These hypotheses help form the probability in terms of a possibility space, which allows a measure holding values between 0 and 1. This is known as the probability measure, to a set of possible outcomes of the sample space.

Probability Density Function

The Probability Density Function (PDF) is the probability function which is represented for the density of a continuous random variable lying between a certain range of values. Probability Density Function explains the normal distribution and how mean and deviation exists. The standard normal distribution is used to create a database or statistics, which are often used in science to represent the real-valued variables, whose distribution is not known.

Probability Terms and Definition

Some of the important probability terms are discussed here:

Applications of Probability

Probability has a wide variety of applications in real life. Some of the common applications which we see in our everyday life while checking the results of the following events:

• Choosing a card from the deck of cards
• Flipping a coin
• Throwing a dice in the air
• Pulling a red ball out of a bucket of red and white balls
• Winning a lucky draw

Other Major Applications of Probability

• It is used for risk assessment and modelling in various industries
• Weather forecasting or prediction of weather changes
• Probability of a team winning in a sport based on players and strength of team
• In the share market, chances of getting the hike of share prices

Problems and Solutions on Probability

Question 1: Find the probability of ‘getting 3 on rolling a die’.

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

Total number of outcomes = n(S) = 6

Let A be the event of getting 3.

Number of favourable outcomes = n(A) = 1

i.e. A  = {3}

Probability, P(A) = n(A)/n(S) = 1/6

Hence, P(getting 3 on rolling a die) = 1/6

Question 2: Draw a random card from a pack of cards. What is the probability that the card drawn is a face card?

A standard deck has 52 cards.

Total number of outcomes = n(S) = 52

Let E be the event of drawing a face card.

Number of favourable events = n(E) = 4 x 3 = 12 (considered Jack, Queen and King only)

Probability, P = Number of Favourable Outcomes/Total Number of Outcomes

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

P(the card drawn is a face card) = 3/13

Question 3: A vessel contains 4 blue balls, 5 red balls and 11 white balls. If three balls are drawn from the vessel at random, what is the probability that the first ball is red, the second ball is blue, and the third ball is white?

The probability to get the first ball is red or the first event is 5/20.

Since we have drawn a ball for the first event to occur, then the number of possibilities left for the second event to occur is 20 – 1 = 19.

Hence, the probability of getting the second ball as blue or the second event is 4/19.

Again with the first and second event occurring, the number of possibilities left for the third event to occur is 19 – 1 = 18.

And the probability of the third ball is white or the third event is 11/18.

Therefore, the probability is 5/20 x 4/19 x 11/18 = 44/1368 = 0.032.

Or we can express it as: P = 3.2%.

Question 4: Two dice are rolled, find the probability that the sum is:

• less than 13

Video Lectures

Introduction.

Solving Probability Questions

Probability Important Topics

Probability Important Questions

Probability Problems

• Two dice are thrown together. Find the probability that the product of the numbers on the top of the dice is: (i) 6 (ii) 12 (iii) 7
• A bag contains 10 red, 5 blue and 7 green balls. A ball is drawn at random. Find the probability of this ball being a (i) red ball (ii) green ball (iii) not a blue ball
• All the jacks, queens and kings are removed from a deck of 52 playing cards. The remaining cards are well shuffled and then one card is drawn at random. Giving ace a value 1 similar value for other cards, find the probability that the card has a value (i) 7 (ii) greater than 7 (iii) less than 7
• A die has its six faces marked 0, 1, 1, 1, 6, 6. Two such dice are thrown together and the total score is recorded. (i) How many different scores are possible? (ii) What is the probability of getting a total of 7?

Frequently Asked Questions (FAQs) on Probability

What is probability give an example, what is the formula of probability, what are the different types of probability, what are the basic rules of probability, what is the complement rule in probability.

In probability, the complement rule states that “the sum of probabilities of an event and its complement should be equal to 1”. If A is an event, then the complement rule is given as: P(A) + P(A’) = 1.

What are the different ways to present the probability value?

The three ways to present the probability values are:

• Decimal or fraction

What does the probability of 0 represent?

The probability of 0 represents that the event will not happen or that it is an impossible event.

What is the sample space for tossing two coins?

The sample space for tossing two coins is: S = {HH, HT, TH, TT}

Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin!

Select the correct answer and click on the “Finish” button Check your score and answers at the end of the quiz

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good basics concepts provide

Great video content. Perfectly explained for examinations. I really like to learn from BYJU’s

Thank you for your best information on probablity

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Well explained

Excellent explanation of probability. One can easily understand about the probability.

I really appreciated your explanations because it’s well understandable Thanks

Love the way you teach

There are 3 boxes Box A contains 10 bulbs out of which 4 are dead box b contains 6 bulbs out of which 1 is dead box c contains 8 bulbs out of which 3 are dead. If a dead bulb is picked at random find the probability that it is from which box?

Probability of selecting a dead bulb from the first box = (1/3) x (4/10) = 4/30 Probability of selecting a dead bulb from the second box = (1/3) x (1/6) = 1/18 Probability of selecting a dead bulb from the third box = (1/3) x (3/8) = 3/24 = 1/8 Total probability = (4/30) + (1/18) + (1/8) = (48 + 20 + 45)360 =113/360

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Statistics and probability

Course: statistics and probability   >   unit 7.

• Intro to theoretical probability

Probability: the basics

• Simple probability: yellow marble
• Simple probability: non-blue marble
• Simple probability
• Intuitive sense of probabilities
• Comparing probabilities
• The Monty Hall problem

• The probability of an event can only be between 0 and 1 and can also be written as a percentage.
• The probability of event A ‍   is often written as P ( A ) ‍   .
• If P ( A ) > P ( B ) ‍   , then event A ‍   has a higher chance of occurring than event B ‍   .
• If P ( A ) = P ( B ) ‍   , then events A ‍   and B ‍    are equally likely to occur.

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Probability

How likely something is to happen.

Many events can't be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability.

Tossing a Coin

When a coin is tossed, there are two possible outcomes:

Heads (H) or Tails (T)

• the probability of the coin landing H is ½
• the probability of the coin landing T is ½

Throwing Dice

When a single die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, 6 .

The probability of any one of them is 1 6

In general:

Probability of an event happening = Number of ways it can happen Total number of outcomes

Example: the chances of rolling a "4" with a die

Number of ways it can happen: 1 (there is only 1 face with a "4" on it)

Total number of outcomes: 6 (there are 6 faces altogether)

So the probability = 1 6

Example: there are 5 marbles in a bag: 4 are blue, and 1 is red. What is the probability that a blue marble gets picked?

Number of ways it can happen: 4 (there are 4 blues)

Total number of outcomes: 5 (there are 5 marbles in total)

So the probability = 4 5 = 0.8

Probability Line

We can show probability on a Probability Line :

Probability is always between 0 and 1

Probability is Just a Guide

Probability does not tell us exactly what will happen, it is just a guide

Example: toss a coin 100 times, how many Heads will come up?

Probability says that heads have a ½ chance, so we can expect 50 Heads .

But when we actually try it we might get 48 heads, or 55 heads ... or anything really, but in most cases it will be a number near 50.

Learn more at Probability Index .

Some words have special meaning in Probability:

Experiment : a repeatable procedure with a set of possible results.

Example: Throwing dice

We can throw the dice again and again, so it is repeatable.

The set of possible results from any single throw is {1, 2, 3, 4, 5, 6}

Outcome: A possible result.

Example: "6" is one of the outcomes of a throw of a die.

Trial: A single performance of an experiment.

Example: I conducted a coin toss experiment. After 4 trials I got these results:

Three trials had the outcome "Head", and one trial had the outcome "Tail"

Sample Space: all the possible outcomes of an experiment.

Example: choosing a card from a deck

There are 52 cards in a deck (not including Jokers)

So the Sample Space is all 52 possible cards : {Ace of Hearts, 2 of Hearts, etc... }

The Sample Space is made up of Sample Points:

Sample Point: just one of the possible outcomes

Example: Deck of Cards

• the 5 of Clubs is a sample point
• the King of Hearts is a sample point

"King" is not a sample point. There are 4 Kings, so that is 4 different sample points.

There are 6 different sample points in that sample space.

Event: one or more outcomes of an experiment

Example Events:

An event can be just one outcome:

• Getting a Tail when tossing a coin
• Rolling a "5"

An event can include more than one outcome:

• Choosing a "King" from a deck of cards (any of the 4 Kings)
• Rolling an "even number" (2, 4 or 6)

Hey, let's use those words, so you get used to them:

Example: Alex wants to see how many times a "double" comes up when throwing 2 dice.

The Sample Space is all possible Outcomes (36 Sample Points):

{1,1} {1,2} {1,3} {1,4} ... ... ... {6,3} {6,4} {6,5} {6,6}

The Event Alex is looking for is a "double", where both dice have the same number. It is made up of these 6 Sample Points :

{1,1} {2,2} {3,3} {4,4} {5,5} and {6,6}

These are Alex's Results:

 After 100 Trials , Alex has 19 "double" Events ... is that close to what you would expect?

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Keyboard Shortcuts

General probability rules.

Rule 1: The probability of an impossible event is zero; the probability of a certain event is one. Therefore, for any event A, the range of possible probabilities is: 0 ≤ P(A) ≤ 1

Rule 2: For S the sample space of all possibilities, P(S) = 1. That is the sum of all the probabilities for all possible events is equal to one. Recall the party affiliation above: if you have to belong to one of the three designated political parties, then the sum of P(R), P(D) and P(I) is equal to one.

Rule 3: For any event A, P(A c ) = 1 - P(A). It follows then that P(A) = 1 - P(A c )

Rule 4 (Addition Rule): This is the probability that either one or both events occur

a. If two events, say A and B, are mutually exclusive - that is A and B have no outcomes in common - then P(A or B) = P(A) + P(B)

b. If two events are NOT mutually exclusive, then P(A or B) = P(A) + P(B) - P(A and B)

Rule 5 (Multiplication Rule): This is the probability that both events occur

a. P(A and B) = P(A) • P(B|A) or P(B)*P(A|B) Note: this straight line symbol, |, does not mean divide! This symbols means "conditional" or "given". For instance P(A|B) means the probability that event A occurs given event B has occurred.

b. If A and B are independent - neither event influences or affects the probability that the other event occurs - then P(A and B) = P(A)*P(B). This particular rule extends to more than two independent events. For example, P(A and B and C) = P(A)*P(B)*P(C)

Rule 6 (Conditional Probability): \(P(A|B)=\frac{P(A \ and \ B)}{P(B)}\) or \(P(B|A)=\frac{P(A \ and \ B)}{P(A)}\)

Syllabus Edition

First teaching 2021

Last exams 2024

Basic Probability ( CIE IGCSE Maths: Extended )

Revision note.

Basic Probability

What is probability.

• Probability describes the likelihood of something happening
• In real-life you might use words such as impossible, unlikely, certain, etc to describe probability
• This means giving it a number between 0 (impossible) and 1 (certain)
• Probabilities can be given as fractions, decimals or percentages

What key words and terminology are used with probability?

• Trials are what we call the repeats of the experiment
• An outcome is a possible result of a trial
• Events are usually denoted with capital letters: A, B, etc
• n( A ) is the number of outcomes that are included in event A
• An event can have one or more than one outcome
• It can be represented as a list or a table
• The probability of event A is denoted as P( A )

How do I calculate basic probabilities?

• The t heoretical probability of an event can be calculated without using an experiment by dividing the number of outcomes of that event by the total number of outcomes
• In some situations, identifying all possible outcomes using a list or a table can help

How do I find missing probabilities?

• If you have a table of probabilities with one missing then you can find it by subtracting the (sum of the) rest from 1
• This can be thought of as not A
• This is denoted  A'

What are mutually exclusive events?

• For example: when rolling a dice the events “getting a prime number” and “getting a 6” are mutually exclusive
• If A and B are mutually exclusive events then to find the probability that A   OR   B occurs you can simply add together the probability of A and the probability of B
• Complementary events are mutually exclusive
• Probabilities can be fractions, decimals or percentages (but nothing else!)
• If no format is indicated in a question then fractions are normally best

Worked example

Emilia is using a spinner that has outcomes and probabilities as shown in the table.

The spinner has an equal chance of landing on blue or red.

Complete the probability table.

The probabilities of all the outcomes should add up to 1 .

1 - 0.2 - 0.1 - 0.4 = 0.3

The probability that it lands on blue or red is 0.3. As the probabilities of blue and red are equal you can halve this to get each probability.

0.3 ÷ 2 = 0.15

Now complete the table.

Find the probability that the spinner lands on green or purple.

As the spinner can not land on green and purple at the same time they are mutually exclusive. This means you can add their probabilities together.

0.1 + 0.4 = 0.5

P(Green or Purple) = 0.5

Find the probability that the spinner does not land on yellow.

The probability of not landing on yellow is equal to 1 minus the probability of landing on yellow.

1 - 0.2 = 0.8

P(Not Yellow) = 0.8

Possibility Diagrams

What is a possibility diagram (sample space).

• In probability, the sample space  means all the possible outcomes
• the letters H, T can be used
• For rolling a six-sided die, the sample space is:  1, 2, 3, 4, 5, 6 
• For example, rolling two six-sided dice and adding their scores
• It would be hard to spot whether you had missed any possibilities
• It would be hard to spot any patterns in the sample space
• Use a possibility diagram instead

• In this case the sample space is:  HHH, HHT, HTH, THH, HTT, THT, TTH, TTT (8 possible outcomes)

How do I use a possibility diagram to calculate probabilities?

• For a fair six-sided die: 1, 2, 3, 4, 5, 6 are all equally likely
• For a fair ( unbiased ) coin: H, T are equally likely
• count the number of outcomes that sum to 7 (there are 6 of them) - this goes in the denominator
• count the number of  those outcomes in which one of the dice shows a 6 (there are two of these, (1,6) and (6,1)) - this goes in the numerator

Some harder questions may not say "by drawing a possibility diagram" so you have to decide to do it on your own.

Two fair six-sided dice are rolled.

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Mathematics class 10 lab manual | 21 lab activities, math assignment ch-14 class x | probability.

a) A face card [Ans 3/13]

b) A red face card [Ans 3/26]

c) A spade card [Ans 1/4]

d) A king of red color [Ans 1/26]

e) A jack of heart [Ans 1/52]

f) An ace card [Ans 1/13]

g) A card of spade or ace [Ans 4/13]

h) Either king or queen [Ans 2/13]

i) Neither king nor queen [Ans 11/13]

j) Neither red nor queen [Ans 6/13]

k) 10 of black suit [Ans 1/26]

a) A black ball [Ans 1/2]

b) A red or a black ball [Ans 11/14]

c) A red white or black ball [Ans 1]

d) Not a red ball [Ans 5/7]

e) Neither white nor black ball [Ans 2/7]

f) Not a white ball [Ans 11/14]

a) Difference of the numbers on the two dice is 2 [Ans 2/9]

b) Sum of numbers on the two dice is 10 [Ans 1/12]

c) Sum of numbers on two dice is more than 9 [Ans 1/6]

d) Six will come up at least once [Ans 11/36]

e) Both are odd number [Ans 1/4]

f) Six as a product. [Ans 1/9]

g) A doublet [Ans 1/6]

h) Getting same number [Ans 1/6]

i) Getting different number [Ans 5/6]

j) A total of 9 or 11 [Ans 1/6]

a) A Leap Year [Ans 2/7]

b) In a non leap year [Ans 1/7]

a) Not red [Ans 5/6]

b) A white ball [Ans 1/3]

a)Two tails   [Ans 3/8]

b) At least two tails   [Ans 1/2]

c) At most two tails   [Ans 7/8]

d) At most 3 heads   [Ans 1]

e) At least 3 heads    [Ans 1/8]

f) Exactly two heads  [Ans 3/8]

a) Exactly one head  [Ans 2/4 = 1/2]

b) Exactly one tail  [Ans 2/4]

c) At least two tails  [Ans 1/4]

d ) A most two tails  [Ans 4/4 = 1]

e) Not less than one head  [Ans 3/4]

f) No tail   [Ans 1/4]

a) A heart [Ans 13/49]

b) A club [Ans 10/49]

c) A king [Ans 3/49]

d) A 10 of heard [Ans 1/49]

a) A prime number [Ans 11/35]

b) A multiple of 7 [Ans 1/7]

c) Multiple of 3 or 5 [Ans 16/35]

a) A consonant [Ans 21/26]

b ) A vowel [Ans 5/26]

a) Letter M or A

b) Letter T

c) Letter E

b) A consonant

a) Will be a multiple of 5 [Ans 3/4]

b) Will be a multiple of 9 [Ans 0]

c) Will end with 7 [Ans 1/4]

d) Does not end with zero [Ans 2/4]

a) Getting odd number

b) Getting a number divisible by 4

c) Getting a number less than 30

d) Getting a number greater than 50

Hint : All possible numbers with sum 12 are: [39, 93, 48, 84, 57, 75, 66]

a) Red or Black

b) Not Green

c) Neither white nor Black

c) white or blue

a) No Head 

b) One Head

c) Two Head 

d) Three Head

a) Probability of sure event…………

b) Probability of impossible event………

c) Range of probability…………

d) Probability of getting sum 13 in single through of a pair of dice………

e) P(E) + P(not E) = …………………..

f) What are equally likely events

a) 1    b) 0      c)  0 ≤ P(E) ≤ 1    d)  0    e)  1

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PAD 2025: Probability, Analysis and Dynamics

Date: 9 - 11 April 2025

Location: University of Bristol

Event type: Conference

Organisers: Márton Balázs (Bristol), Edward Crane (Bristol), Asma Hassannezhad (Bristol), Kevin Hughes (Bristol), Jessica Jay (Bristol), Ben Krause (Bristol), John Mackay (Bristol), Jens Marklof (Bristol)

Probability, analysis and dynamics are three central areas of mathematics in which the UK has a leading position. It is difficult to say exactly where the boundaries between the three fields lie; they have many tools in common and problems in one of the fields are often motivated by results or problems in another. All three topics have physics as an underlying source of questions. Despite this, there are relatively few opportunities for researchers in these fields to interact with each other. This conference gives mathematicians the opportunity to interact with the international speakers who are setting the research agendas for their subjects.

CMI Enhancement and Partnership Program

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