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Probability 

Probability is a branch of mathematics that deals with calculating the likelihood of given event’s occurrence, which is expressed as a number between 0 and 1. We usually predict about many events based on certain parameters. 
 

Ex: Getting a head, when a coin is tossed. Getting a odd number, when a die is rolled. The better we know about the parameters related to an event result then the result predicted accuracy is better.
 

Mathematically, 

Probability of happening an event 

 All possibilities related to an event are known as total possible number of possible outcomes. 
 The possibilities favourable to an event are known as favourable outcomes.

Basic terms related to probability

Experiment: An action where the result is uncertain although the all possible outcomes known in advance. 

Ex: Throwing a die. 

Sample space: All possible outcomes of an experiment is called as sample space. 

Ex: If we throw a die, then sample space S = {1, 2, 3, 4, 5, 6}

Event: Single result of an experiment. 

Ex: Getting a tail is an event related to tossing a coin.

 Types of events:

Certain event: A certain event is sure to occur probability of certain event is 1.

Ex: In a tossing a coin once getting a tail or head. 
 

Impossible event: An impossible event is no chance of occurring. 
Probability of impossible event is 0.

Ex: Getting a number greater than 6 when a die is rolled.

Equally likely events: If probability of occurrence of each event in an experiment is same then the events are said to be equally likely events.

Ex: When a die is rolled the possible outcomes of getting an odd number = possible outcomes of getting an even number = 3

Complementary events: The complement of an event A is the set of outcomes in the sample space. That are not included in the outcomes of event A. 
P(A − ) = 1 − P(A)
        Mutually exclusive events if the occurrence of one event excludes the occurrence of another event, such events are mutually exclusive event i.e two events don’t have any common point
Ex: If S = {1, 2, 3, 4, 5, 6} 
A and B are two events such that A consist of number less than 3 and B consists of numbers greater than 4.
 Mutually Exhaustive events: A set of events is called exhaustive if all the events together consume the entire sample space.

Ex: If S = {head, tail}

x: If S = {head, tail} In the experiment of tossing coin the event of getting head occur, the event of tail occur are mutually exhaustive because both events covers the sample space. 
 

Based on coins 

1. A coin is tossed twice, then find the probability the head is obtained two times.
Sol: When a coin is tossed twice, then possible outcomes are 
n(S) = {HH, HT, TH, TT} = 4 
Favourable outcomes are (head is obtained two times) n(P) = {HH} = 1 

Note: If it is two coins we can write the sample space and favourable outcomes easily, but more than 3 coins it is not that much of easy to write total outcomes and favourable outcomes. 
So here we can use a small technique,

 When a coin is tossed n times, then possible outcomes are 
 When a coin is tossed 2 times, then possible outcomes are n(S) = 22
 Favourable outcomes are (head is obtained two times) n(P) = 2C2 

2. A coin is tossed 3 times, then find the probability the head is obtained 1 time. 
Sol: When a coin is tossed 3 times, then possible outcomes are n(S) = 23 = 8 
Favourable outcomes are (head is obtained 1 time) n(P) = 3C1 = 3 

Based on dice

1. A single 6 sided die is rolled, then find the probability of getting an even number. 
Sol: Total possible outcomes are n(S) = {1, 2, 3, 4, 5, 6} = 6 
Favourable outcomes are n(P) = {2, 4, 6} = 3 

2. A single 6 sided die is rolled, then find the probability of getting 3 or 5. 
Sol: Total possible outcomes are {1, 2, 3, 4, 5, 6} = 6
Favourable outcomes of getting 3 = {3} = 1,
then probability of getting 3 =
Favourable outcomes of getting 5 = {5} = 1,
then probability of getting 5 =    
Probability of getting 3 or 5 =  
 = (both events are mutually exclusive 3 means there is no point of intersection) 

3. When two dice are rolled, find the probability the sum of the numbers on the dice is 8. 
Sol: When n dice are rolled, then total possible outcomes are = 6n When 2 dice are rolled, then total possible outcomes are = 6
(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) 
(2, 1).......................................... 
(3, 1)................................................. 
(4, 1)................................................. 
(5, 1)................................................. 
(6, 1).................................................

Based on marbles

 A glass jar contains 1 red, 3 green, 2 blue and 4 yellow marbles. If a single marble is chosen at random from the jar, what is the probability that it is yellow or green?
Sol: We can choose 1 marble from total marbles Total outcomes (total marbles) 
                 = 1 + 3 +  2 + 4 3
                 = 10C
Favourable outcomes of getting green marble   
                                                      = 3C1        = 3

Probability of getting yellow marble = 

Favourable outcomes of getting yellow marble   = 4C1 = 4

Probability of getting green marble = 

Probability of getting yellow marble or green marble 

                 =   


Note: If we write total outcomes and favourable outcomes all the time. It consumes a lot of time so here we can use a small technique (for 2 dice rolled).

Note: If the sum is more than 7 (8 or 9 or 10 or 11 or 12), just we can subtract sum from 14 resulting number probability is the answer

Ex: If the sum is 8 then subtract 8 from 14 = (14 − 8) = 6

Based on playing cards

       Playing cards (52) 
Black (26)                       Red (26) 
Clubs (13)                        Spades (13) 
Hearts (13)                     Diamonds(13) 

 In every suit of clubs or spades or hearts or diamonds ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king (13 cards) 

 There are 13 cards of each suit clubs or spades or hearts or diamonds. 
 There are 4 ace, 4 jacks, 4 queens, 4 kings. 
 There are 12 face cards (king, queen, jack) 
 There are 16 honour cards (Ace, king, queen, jack

Example: A single card is chosen at a random from standard deck of playing 52 cards.

1. What is the probability that choosing a king? 
Sol: Total outcomes (total cards) = 52C1 = 52 (one card is selected from total cards) Favourable outcomes (total kings) = 4C1 = 4 (one card is selected from total kings)


 

2. What is the probability that choosing a black card? 
Sol: Total outcomes (total cards) = 52C1 = 52 (one card is selected from total cards) Favourable outcomes (total black cards) = 13C1 = 13 (one card is selected from total black cards)

favourable outcomes  (total outcomes)
     
 

3. What is the probability that choosing a face card? 
Sol: Total outcomes (total cards) = 52C1 = 52 (one card is selected from total cards)
Favourable outcomes (total face cards) = 12C1 = 12 (one card is selected from total face cards) 

Posted Date : 29-11-2021

గమనిక : ప్రతిభ.ఈనాడు.నెట్‌లో కనిపించే వ్యాపార ప్రకటనలు వివిధ దేశాల్లోని వ్యాపారులు, సంస్థల నుంచి వస్తాయి. మరి కొన్ని ప్రకటనలు పాఠకుల అభిరుచి మేరకు కృత్రిమ మేధస్సు సాంకేతికత సాయంతో ప్రదర్శితమవుతుంటాయి. ఆ ప్రకటనల్లోని ఉత్పత్తులను లేదా సేవలను పాఠకులు స్వయంగా విచారించుకొని, జాగ్రత్తగా పరిశీలించి కొనుక్కోవాలి లేదా వినియోగించుకోవాలి. వాటి నాణ్యత లేదా లోపాలతో ఈనాడు యాజమాన్యానికి ఎలాంటి సంబంధం లేదు. ఈ విషయంలో ఉత్తర ప్రత్యుత్తరాలకు, ఈ-మెయిల్స్ కి, ఇంకా ఇతర రూపాల్లో సమాచార మార్పిడికి తావు లేదు. ఫిర్యాదులు స్వీకరించడం కుదరదు. పాఠకులు గమనించి, సహకరించాలని మనవి.

 

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