Probability - Definitions, Formulas, Problems

By: Admin | Posted on: Apr 22, 2020

Probability 

Today we will cover Probability, knowing the probability definitions, tips, and important tips or shortcuts is very helpful to crack this topic. Probability is an important topic for most of the competitive exams. Probability problems are an important part of the QA section of most competitive exams like SBI, IBPS, PO/Clerk, LIC-AAO, SSC, etc. So it becomes relevant to know the right technique of solving these questions. Let us start with our topic Probability.

 

Probability Definition and its purpose?

Probability means chances. Probability is the measure of finding how likely an event P(E), is of happening of an event. It tells the purpose of how likely the event will happen or not. Situations in which each outcome is equally likely then we can find the probability of an event using probability formula. Probability is a chance of happening something or we can use a word known as a prediction.

For example: what is the probability of an event such that whether it will rain or not

If the probability that an event will occur is P(E), then the probability that the event will not occur is 1 – P(E). if the probability that one event will occur is X and the independent probability that another event will occur is Y then the probability that both events will occur is XY.

 

Probability Definition:

a) An experiment is a situation, which involves chance or probability that leads to results known as outcomes.

b) An outcome for an event is the result of an experiment.

c) An event is one or more outcomes of a random experiment.

d) Probability is the measure of how likely an event is.

With a good understanding of probability definition, one can easily crack the complex problems related to probability.

Some Basic Terms and Concepts of Probability

How you can learn probability in math?

(i) An experiment

An action or operation for an event resulting in two or more outcomes is called an experiment.

(a) Tossing a coin is an experiment. The two possible outcomes for this experiment are head or tail.

(b) Drawing a card from a pack of 52 cards is an experiment. There are 52 possible outcomes.

Trail: Performing a random experiment for an event is called a Trail.

(ii) Sample space

The set of all possible outcomes of while performing any experiment is called the sample space, denoted by S. Elements of S is called a sample point.

e.g.. (a) In the experiment of tossing of a coin, the sample space has two points corresponding to head (H) and Tail (T) S (H, T)

(b) When we throw a die, then any one of the numbers 1, 2,3,4,5 and 6 will come up. The sample space S(for dice) = {1, 2, 3, 4, 5, 6)

(iii) Event: Any subset of sample space is an event.

eg. (a) If the experiment is done throwing a die which = has faces numbered 1 to 6, then

X= (1, 2, 3, 4, 5, 6).

Then. A = (1,3,5), B (2, 4, 6), the null set, and S itself are some events with respect to S.

What are the null and certain events?

Null event: The null set is called the impossible event. Getting 8 when a die is thrown is called an impossible event.

The entire sample space is known as a certain event.

(iv) Simple event: An event is called a simple event if it is a singleton subset of the sample space S.

eg. (a) When a coin is tossed, sample space S (H, T)

Let A = (H) = the event of the occurrence of head.

and B = (T) = the event of the occurrence of the tail.

Here A and B are simple events.

(b) When a die is thrown, sample space S = (1, 2, 3, 4, 5, 6)

 Let A = {5} = the event of occurrence of 5.

B = {2} = the event of occurrences of 2.

Here A and B are simple events.

(v) Compound events it is the joint occurrence of two or more simple events.

Example: The event of at least one head appears when two fair coins are tossed is a compound event ie. A = (HT, TH, HH)

(vi)Equally likely events: The number of simple events is said to be equally likely if there is no reason for one event to occur in preference to any other event.

Example: While drawing Playing cards from a well-shuffled deck of cards there are 52 outcomes as the cards, which are equally likely.

(vii) Exhaustive events: All the possible outcomes taken together in which an experiment can result are said to be exhaustive

Example:  A card is drawn from a well-shuffled pack of 52 cards.

(i) The following events are exhaustive.

(a) The card is black. (b) The card is red.

(ii) The following events are not exhaustive.

(a) The card is the heart. (b) The card is a diamond.

A and B are exhaustive events of the sample space S, then A U B=S

(vii) Mutually exclusive events: If two events cannot occur simultaneously, then they are mutually exclusive.

If A and B are mutually exclusive, then we will write them as A∩B=∅

e.g. In drawing a card from a well-shuffled pack, the following events A = the card is a spade; B = the card is a heart are mutually exclusive.

  • In a single toss of a coin, either the head or the tail will appear and not both.
  • In a playing of a cubic dice, either an odd number or an even number will turn up and not both.

The following events are not mutually exclusive

(a) The card is a heart. (b) The card is a king.

The card can be king of heart.

viii)  Complementary events: Let A be an event of a random experiment and S the sample space. All the other outcome which are not in A belong to the subset S–A. The event S – Ais called the complement of E.

ix) Sure Event: Since S is a subset of S, S itself is an event and S is called a sure or certain event.

x) Impossible event: Let F be an event of getting more than two heads in tossing two coins simultaneously. F = {} = Ø. So F is an impossible event. Therefore, A sample is a sure event. An empty set Ø is an impossible event.

Before going through the various examples. Let us see important formulae which will help to solve various questions in the examination.

 

Important Probability Formulae and Rules

  1. P(E)= Number of favorable outcome/Total number of outcome
  2. P(E) + P(Ē) = 1
  3. For any Event E, probability lies between 0 to 1
  4. It involves multiple events
  5. It also involves multiplication rules and additional rules for two events defined on the sample space.

Now the question arises that what are some common cases or situations that we study in probability. So here we have the following cases on which most problems are asked.

 

Probability in Real life

Probability is the calculation of the occurrence of the event happening or not. There are many real-life scenarios where probability concepts are applied. Below are some common probability applications in real life.

Cases in Probability

  1. Dice Problems
  2. Coins
  3. Cards
  4. Ball Or Boxes

How do you calculate the probabilities of the various events?

1. Dice:

The dice used here is the one we used to play ‘Ludo, Snake, and Ladders’.

A dice has numbers 1, 2, 3, 4, 5, and 6 are written over its six faces as shown below.

When a dice is thrown the number that appears on the upper face is an event. 

The Problems based on dice are mainly of two types

(I) When Only One dice is thrown once and sample spaces :


In such cases, the number rolling on playing dice is either one of them {1,2,3,4,5,6}.
Here assuming, the concerned event of rolling out ‘1’ is 1 only because 1 is written only one face. 
hence, the Probability of rolling number 1 is given by  = 1/6

Similarly, Probability of rolling number 2 (or any number between 3 to 6)

Probability of favorable outcome is 1

Total number of possible outcome is 6

Hence the probability of an event is given by   = 1/6

 

Types of questions asked:
 

1. What is the probability of getting an even number on the rolling a dice?

Now, considering the event should have an even number which is 2, 4 and 6 on the dice
So, the probability of Concerned Event( Favorable outcome) = 3
Total number of possible out came= 6
So, the Probability of the event is given by = 3/6 = ½

2. What is the probability of getting an Odd number on rolling a dice?

Now, considering the event should have an even number which is 1, 3 and 5 on the dice
So, the probability of Concerned Event( Favorable outcome) = 3
Total number of possible out came= 6
So, the probability of the event is given by = 3/6 = 1/2

3. What is the probability of getting 6 on rolling a dice?

Now, considering the event should have a 6 on the dice
Hence, the probability of Concerned Event (Favorable outcome) = 1
Total number of possible outcome= 6
So, the Probability of the event is given by = /6 = 1/2


4. What is the probability of getting a combination of ‘2’ and ‘6’ on throwing two dices?

Solutions – Now the concerned event should have number ‘2’ and ‘6’ So concerned events = 2{(2,6) and (5,6)}
Total events = 36
⇒ Probability of the event is given by = 2/36 = 1/18

 

5. What is the probability of getting a sum of ‘9’ on rolling a dice twice?

Solution – Now, the concerned events should have a sum of 9 i.e. Two dice are required. On which adding the numbers on the dice must be 9.

Number on dice 1 + Number on dice 2 = 9
these cases will fulfill condition: (4, 5); (5, 4) and (6, 3), (3, 6)
Total number of possible outcome = 4
and, we know Total Events ‘36’
⇒ Probability of the event is given by = 4/36 = 1/9

2. Coins: 

The coin is a currency token, which has two faces, one is head and the other is one tail. So, when tossing a coin in the air and when it lands, it might have either a head or tail faces. Coin questions can be classified into three types of categories as shown below:

I. One Coin once:
When a coin is tossed is only once then there can be two possible outcomes either it will be ahead or a tail. In such cases, the total event is 2

Now let us see some event-based questions on the Coin problems.


Question: Consider an event of tossing a coin. What is the probability of getting ahead?
Solutions Concerned event = 1(One head)
Total Event = 2
⇒ P(E) = 1/2

Question: Consider an event of tossing a coin. What is the probability of getting a Tail?
Solutions Concerned event = 1(One Tail)
Total Event = 2
⇒ P(E) = ½

II. Two Coins or One Coin Twice:
When two coins are tossed together or one coin is tossed two times then the following outcomes can be obtained:

Here, ‘H’ = Head; ‘T’ = Tail.

(H, T) shows that on coin 1 it’s Head while on coin 2 its Tail. Here, we have

Total Event = 4

Question: What is the probability of getting at most one head-on tossing a coin?
Solutions – At most one head means there can be either 0 head or 1 head.
So, Concerned Event = 3 {(H, T) (T, H) (T, T)}
Total Events = 4
⇒ P(E) = 3/4

 

3. Cards:

There is a total of 52 cards in a deck in which we have four kinds of the symbol used in playing cards. In each set, we have 13 cards. The Description of different symbols are as below:


i) Spade ⇒Black in color (13 in number)♠
ii) Club(Clove)Black in color (13 in number)♣
iii) Heart ⇒Redin color (13 in number)♥
iv) DiamondsRedin color (13 in number)♦

Each of these 4 variants has 13 cards each as a number from 2, 3, 4,…..10 and Jack, Queen, King and Ace(1). There are
1) 26 red cards and 26 black cards = 52 Cards
2) In red cards Heart & Diamonds each of 2, 3 …. 10 andJack, Queen, King, and Ace.

3) Similarly In black cards Spade & Clove each of 2, 3 …. 10 and, Jack, Queen, King, and Ace.

So, in total there are 13 × 4 = 52 cards.

Types of questions asked:

1. One card is drawn:
In such types of questions, a card is drawn from a well-shuffled deck of cards. Here, the Total Events = 52

Question: What is the probability of getting a King of diamond or Queen of Heart in one draw?

Solutions – Here, ‘King of diamond or Queen of Heart’ means that either the card can be the diamond King or Heart Queen. Clearly, there is only one King of diamond and only one Queen of Heart.
So, Concerned Event = 2
Total Event = 52
⇒P(E) = 2/52 = 1/26

 

2. What is the probability of getting a RED ace in one draw?

Solution:

P(A)=favorable outcome =Total  of Red Ace = 2

P(B) =total number of outcomes =Total number of cards =52

P(E) =Probability of getting a Red ace =2/52

 

4. Ball/Stones/Marble in (Boxes, Bags)

In such questions, a bag or box contains certain Ball/Stones/Marble and some of them (are) drawn.it may be randomly or it  may behave some condition

I. One drawn:

Question  – In a box, there are 8 Black, 7 blue and 6 pink marble. One marble is picked up randomly. What is the probability that marble is neither black nor green?

Solutions – Total Events = 8 (Black) + 7 (blue) + 6 (Pink) = 21
Since the selected marble has to be neither black nor Pink then it’d be Blue and blue marble are 7.

So, P(E) = 7 ⇒ P(E) = 7/21 = 1/3 

II. More than One drawn without replacement:

Question – A bag contains 10 red or 10 blue marbles. The probability of drawing two marbles of the same colors.

Solution – Total Events (as 2 marbles are drawn) = 20C2
The marbles drawn can either be both Red color or both blue color and for ‘OR’ event we add up the two events.
So, P(E) = 10C2 (if Red) + 10C2 (if blue)
⇒ P(E) = (10C2 + 10C2)÷ 20C2 = 10 x 9 + 10 x 9 / 20 x 19 = 9 / 19

 

Question – A bag contains 10 Red and 10 blue marbles. The probability of drawing two marbles of the same colors.

Solution – Total Events (as 2 marbles are drawn) = 20C2
The balls drawn can be both Red color or both blue color. For ‘And’ event we can multiply both events. So, We have two cases here,

The first case – Probability of getting both marbles as blue –

So, P(E) = 10C0 (if red) x 10C2 (if blue)

⇒P(E) = (10C0 x 10C2) ÷ 20C2 = (1 x 10 x 9) / (20 x 19) = 9 / 38

The second case – Probability of getting both marbles as red –

So, P(E) = 10C2 (if black) x 10C0 (if blue)

⇒P(E) = (10C2 x 10C0)÷ 20C2 = 10 x 9 x 1 / 20 x 19 = 9 / 38

 

 Other Miscellaneous questions:


Q 1. Four people are to be chosen at random from a group of 3 men, 2 women and 4 children. What is the probability of selecting exactly two of them being children?

Solution – Total People in the group = 3 men + 2women + 4children = 9
Since, we have to select 4 people from 9 so Total Event = 9C4
And, since we have to select 2 of children so these 2 person have to be from 4 children so Concerned Event for children= 4C2
And, other 2 people will be from either Men or Women (3+2 = 5), their Concerned event = 5C2
⇒P(E) (for all 4 people) = 4C2 ×5C2
⇒ P(E) = (4C2 ×5C2) ÷ 9C4 = (4 x 3 x 5 x 4) x (2 x3) / 9 x 8 x 7 x 6 = 10 / 21 

 

Q. 2. Ravi and Ashwin give exam. Chance of Ravi’s selection is 1/7 and of Ashwin’s selection is 1/5. Find probability of only one of them is selected.

Solution – P( Ravi selecting) = 1/7
P (Ravi not selecting) = 1 – 1/7 = 6/7
P( Ashwin selecting) = 1/5
P (Ashwin not selecting) = 1 – 1/5 = 4/5
Probability of only one of them is selectedP(E) = P( Ravi selecting)× P (Ashwin not selecting) + P( Ashwinselecting)× P (Ravinot selecting) =  (1/7 x 4/5) + (1/5 x 6/7) = 10/35 = 2/7.

 

Try some more questions on our online Quiz Section.

Drop your Queries and feedback in our Comment box.

Author: Ashwani Kumar


Tags : A short note on probability definitions probability formulas and probability examples Probability for competitive exams

16-Apr-2021

MyronPlaup

dating sites free no registration free adult personals site

18-Apr-2021

RolandFiree

lasix for sale http://lasixst.com/# furosemide generic lasix

02-May-2021

JavierFlafe

what is tinder , browse tinder for free tinder dating app

04-May-2021

Zbbyzb

stromectol nz - ivermectin 2mg ivermectin where to buy for humans

05-May-2021

Zfisjk

ivermectin 3mg pill - stromectol otc buy ivermectin cream

07-May-2021

Lfifah

ivermectin 4000 mcg - stromectol how much it cost ivermectin cost

09-May-2021

Edfjst

ivermectin 2ml - oral ivermectin cost ivermectin 2mg online

10-May-2021

JavierFlafe

browse tinder for free , tinder dating app what is tinder

12-May-2021

BLUTUANDA

cialis on line


13-May-2021

JavierFlafe

tinder date , tider http://tinderentrar.com/

14-May-2021

Ijgzde

no script ed pills for drunks - effective over the counter ed pills are any are there any over-the-counter ed pills

16-May-2021

Ckgacn

elator to most ed pills - pe and ed pills medication search

18-May-2021

Oswqbf

viagra hoe te gebruiken - wat is het verschil tussen kamagra en sildenafil sildenafil bestellen morgen in huis

20-May-2021

Qadwwj

precio de viagra en cipro 25mg price - farmacia online viagra viagra 50 mg precio

22-May-2021

Mthysx

bactrim discount coupon - bactrim 160 800 mg bactrim online no prescrition

24-May-2021

Ggphsy

cialis 100mg cost - cialis professional from india cialis drug

26-May-2021

Uyhpbc

cost of cialis generic - difference between cialis super active cialis professional buy made in usa cialis online

28-May-2021

Kmlpum

cymbalta 10mg - buy cymbalta 60 mg order cymbalta online

30-May-2021

Hvkfbt

generic levothyroxine - levothyroxine 88 mg levothyroxine 135 mcg

01-Jun-2021

Mdbvip

amoxicillin sleep - amoxicillin pill generic amoxicillin cost

03-Jun-2021

Jydzhi

generic for vardenafil - vardenafil generic buy vardenafil from canada

03-Jun-2021

tadalafil gel

tadalafil 30 mg https://nextadalafil.com/ buy tadalafil order tadalafil

03-Jun-2021

order tadalafil

tadalafil dosage https://pulmoprestadalafil.com/ tadalafil pills 20mg 40 mg tadalafil

03-Jun-2021

generic tadalafil united states

tadalafil 40 https://extratadalafill.com/ tadalafil 40 mg from india generic tadalafil

03-Jun-2021

tadalafil 40

tadalafil 40 mg daily https://superactivetadalafil.com/ buy tadalafil us buy tadalafil us

03-Jun-2021

buy tadalafil

buy tadalafil https://tadalafilgenc.com/ tadalafil dosage tadalafil max dose

04-Jun-2021

Djyhrd

compra vardenafil online - vardenafil for sale online buy vardenafil in india

06-Jun-2021

tadalafil pills

tadalafil 60 mg for sale https://tadalafilgenc.com/ tadalafil dosage tadalis sx

06-Jun-2021

Kgfefj

canadian pharmacy no prescription - canadian pharmacy no scripts prescription drugs online

07-Jun-2021

buy tadalafil us

tadalafil 40 mg from india https://pulmoprestadalafil.com/ tadalafil pills tadalafil daily use

08-Jun-2021

Uimbun

cialis prescription usa - where to buy cialis online in canada cialis europe online

10-Jun-2021

Oesdhl

sildenafil online buy india - cheap viagra generic best price 007 viagra

12-Jun-2021

Oevqkh

vardenafil price - levitra 10mg vardenafil side effects in men

14-Jun-2021

Jboqat

vardenafil prices online - vardenafil 120 pills vardenafil hcl

15-Jun-2021

BLUTUANDA

generic cialis


17-Jun-2021



19-Jun-2021

Nltgaa

cialis buy online - cheap cialis without prescription cialis daily cheap

21-Jun-2021

Pdurks

prednisone 5mg online - prednisone 50 mg tablet prednisone 40mg

23-Jun-2021

Vhxryh

vardenafil coupon - buy vardenafil uk how to buy vardenafil online

25-Jun-2021

Zioqlb

lyrica 125 mg - buy lyrica online uk lyrica without rx

28-Jun-2021

Iuvmey

buy zithromax - buy zithromax pills can you buy zithromax online

29-Jun-2021

Ztzsss

buy prednisolone online uk - prednisolone 5mg where can i buy prednisolone tablets

01-Jul-2021

Gahqyp

stromectol tab price - buy oral ivermectin ivermectin price comparison

02-Jul-2021

Dtuzkm

buy real viagra uk - sildenafil best price uk canadian pharmacy viagra 150 mg

04-Jul-2021

Evwxpt

furosemide lasix - doxycycline order online canada buy doxycycline united states

07-Jul-2021

Jtxcgg

dapoxetine pills - gnpriligy.com cialis online

08-Jul-2021

Anxuak

best ed drug - medications for ed compare ed drugs

10-Jul-2021

Ykliqa

purchase prednisone from india - prednisone price south africa deltasone coupon

12-Jul-2021

Kkfvjr

buy modafinil - modafinil and alcohol modafinil buy

13-Jul-2021

Gitgdk

buy accutane 5mg - can you buy accutane online uk buy 40mg accutane online

14-Jul-2021

Cdmlhh

amoxil USA - amoxicillin for dogs without vet prescription amoxicilin without prescription

15-Jul-2021

Ejxfgc

brand vardenafil vs generic - buy vardenafil online europe best price for vardenafil online

16-Jul-2021

Expcib

ivermectin 50 mg - stromectol buy uk ivermectin for humans for sale

17-Jul-2021

Ceduvp

discount cialis tablets - tadalafil generic where to buy generic cialis

18-Jul-2021

Ynzlgy

ivermectin 9 mg tablet - purchase oral ivermectin ivermectin and covid

19-Jul-2021

Xjipxv

accutane australia - accutane pills price in india accutane price in india

20-Jul-2021

Bkrexn

lyrica buy online uk - vyvanse canadian pharmacy viagra online canadian pharmacy

21-Jul-2021

Ysssvx

buy amoxicilin 500 mg mexico - amopills.com generic amoxil

22-Jul-2021

Xkugcf

buy generic viagra - viagra online pharmacies order viagra online with paypal

22-Jul-2021

BLUTUANDA

best price cialis

23-Jul-2021

Fmozfc

cialis brand for sale - can i purchase cialis over the counter generic tadalafil 2018

24-Jul-2021

Vkjcxu

ivermectin iv - ivermectin 6mg tablet for lice ivermectin 3mg for lice

25-Jul-2021

Giuirw

prednisone 60mg daily - can you buy prednisone online prednisone 3 tablets daily

27-Jul-2021

Igitts

provigil for sale - modafinil 200 mg modafinil and caffeine

29-Jul-2021

Okasyd

generic zithromax 100mg - azithromycin zithromax buy zithromax 500mg online

Leave a Comment