Probability for Business Management Simplified
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PROBABILITY FOR BUSINESS MANAGEMENT SIMPLIFIED
by : DR. T.K. JAIN AFTERSCHO☺OL centre for social entrepreneurship sivakamu veterinary hospital road bikaner 334001 rajasthan, india for PGPSE participants PGPSE : FREE FOR ALL, OPEN FOR ALL mobile : 91+9414430763
5 DECEMBER 09 AFTERSCHOOOL centre for social entrepreneurship bikaner 1
WHAT IS EXCLUSIVE EVENT?
WHEN RESULTS FROM SOME EXPERIMENT ARE SUCH THAT IF ONE OUTCOME (EVENT) HAPPENS, OTHERS ARE IMPOSSIBLE, THEN THEY ARE CALLED EXCLUSIVE EVENTS. FOR EXAMPLE, IF YOU TOSS A COIN, YOU CAN GET EITHER HEAD OR TAIL – NOT BOTH - THESE ARE EXCLUSIVE EVENTS
5 DECEMBER 09 AFTERSCHOOOL centre for social entrepreneurship bikaner 2
WHAT IS PROBABILITY ?
FINDING THE POSSIBILTY OF AN EVENT OUT OF MANY EQUALLY LIKELY EVENTS IS CALLED PROBABILITY THUS IF YOU THROW A COIN, THE PROBABILITY OF GETTING HEAD IS ½ BECAUSE THERE ARE 2 EQUALLY LIKELY EVENTS AND ONE HEAD.
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What is exhaustive list of events?
All the possibilities are called exhaustive list. For example : coin has two possibilities : head and tail – thus these two are exhaustive - there is no other possibilitiy
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Q : what is the probability of getting 2 heads out of 3 tosses of coins?
First : list down all the possibilities : H= Head, T=tail we may have all heads : HHH, we may have all tail : TTT : 1. HHH 2 TTT 3 HTH 4 THT 5 HHT 6 HTT 7 TTH 8 THH in simple terms : 2^3 = 8 possibilities : (the formula is options ^ number of trials) out of these the cases of 2 heads are : 3/8 answer
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What is the possibility of getting a difference of 2 when you toss a dice twice?
Let us list down all the possibilities : (1,1) (1,2)(13)(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), (6,1),(6,2),(6,3),(6,4),(6,5),(6,6) there are total 36 possibilities out of which, 8 cases involve difference of 2, so answer : 8/36
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What is the chance of picking up a spade or an ace from a pack of cards?
There are 13 spade cards. There are 4 ace cards. Thus there are 13+4 cards = 17 out of which one ace+spade is included thus we have 16 out of 52 possibilities : 16/52 answer
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What is the probability that a committee of 7 persons has 5 men and 2 women, it is to be formed out of 8 men and 5 ladies
We have to pick up 7 persons out of (8+5) = 13 members so the total possibilities are : 13C7 = 13! / (7! * (13-7)!) =13*12*11*10*9*8*7*6*5*4*3*2*1/ (7*6*5*4*3*2*1*6*5*4*3*2*1) =1716 methods of selecting 5 men and 2 women : 8C5 * 5C2 = (8*7*6/3*2*1) (5*4/2*1) = 560 answer = 560/1716 answer
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A number is picked up out of 25 natural numbers, what is the probability that it is divisible by 4?
Out of 25 number, there are 6 numbers divisible by 4 : 4,8,12,16,20,24 thus the probability is : 6/25 answer
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P (A-B) = 1/5 and P(A)=1/3 and P(B)=1/2, what is the probability that out of A and B, only B will happen?
P(A-B) = P(A) (1/3)– (P(A intersection B) = 1/5 thus P(A intersection B)= 1/3-1/5= 2/15 we want P (B-A) = P(B)-P(A intersection B) =1/2 – 2/15 = 11/30 ans.
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A hits 5 out of 9 correct shots and B hits 6 out of 11 correct, what is the probability that when they try they are able to hit correct?
First find out the possibility that target is NOT hit at all : A not hitting : (1-5/9) = 4/9 A = 4/9 and B = 5/11 A *B = 4/9 * 5/11 = 20/99 thus they will hit : 1-20/99 = 79/99 answer
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8 balls are distributed at random among 3 boxes, what is the probability that the first box will get 3 balls?
8 balls can be distributed to 3 boxes in 3^8 manners. We want first box to have 3 balls, so 8C3, now we have 5 balls left out which have to be distributed to remaining 2 boxes : 2^5 answer : (8C3*2^5)/3^8 = 1792/6561 answer
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Odds in favour of an event is 2:3 and odds against another event is 3:7 ,what is the probability that at least one of these will happen
First event : happening : 2/5 not happening : 3/5, 2nd event happen : 7/10 and not happen : 3/10 Find the probability that none of these happen : 3/5 * 3/10 = 9/50 at least one of them will happen is : 1-9/5 =41/50 answer
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Two balls are picked up from a bag containing 5 red and 7 blue balls at random, what is the probability that the balls are of 2 different colours?
1 out of 5 = 5C1 and 1 out of 7 = 7C1 total : 12C2 solution : (5*7)/66 =35/66 answer
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What is the probability of getting at least 7 in a single cast with 2 dice?
Let us list down all the possibilities : (1,1) (1,2)(13)(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), (6,1),(6,2),(6,3),(6,4),(6,5),(6,6) out of these : 16 cases have 7 or more than 7 as total so answer is : 21/36 or 7/12 answer
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What is the probabilty of having at least one 6 from 3 throws of a dice?
Having 0 six : 5/6*5/6*5/6* =125/216 having at least 1 six = 95/216
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If 2 letters are taken at random from the word home, what is the probabilty that none of the letters would be vowels?
Vowels are 2 and constants are 2. so : select 2 out of 2 divided by 2 out of 4. 2C2 / 4C2 2C2 = 1 and 4C2 = 6 so answer : 1/6 answer
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A,B,C are 3 mutually independent with probabilities .3, .2 and .4 respectively, what is P(AintersectionBintersectionC) ?
Aintersection B intersection C = .3*.2*.4 = .024 answer
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There are 10 balls numbered from 1 to 10 in a box if one of them is selected at random, what is the probability that the number printed on the ball would be an odd number greater than 4?
We have 5,7,9 as three numbers which are less than 10 but are more than 4 and are odd numbers. So the probability is 3/10 answer
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The wages of 8 workers in $ : 50,62,40,70,45,56,32,45, one worker is selected randomly, what is the probability that his wages would be lower than averge wages?
Total of all : 400, so divided by 8 gives 50. thus we have 4 values less than 50 so the answer is : 4/8 or 50% answer
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If an unbiased coin is tossed 3 times, what is the probability of getting more than 1 head?
List down all the possibilities : TOTAL NUMBER : 2^3 = 8 HHT,HHH,HTH,THH,TTT,TTH,THT,HTT = 4/8 or ½ answer
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If two dice are rolled, what is the probability of getting neither 6 nor 9 as total ?
Let us list down all the possibilities : (1,1) (1,2)(13)(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), (6,1),(6,2),(6,3),(6,4),(6,5),(6,6) total of 6 : 5 cases, total of 9 : 4 cases remaining cases : 27 probability : 27/36 = .75 answer
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What is the probability that 4 children selected at random have different birthdates ?
There are 365 days in a year. So the probabilities are : for the first date – we have 365 options, for 2nd date, we have 364 days (one date is reduced), for 3rd person, we have 343 days and so on. 365/365*364/365*363/365*362/365
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A box contains 5 white and 7 black balls, two successive draws of ball are made with replacements, what is the probability n d that the first ball is white and 2 is black ball?
First white : 5C1 / 12C1 2nd black : 7C1/ 12C1 =5/12 * 7/12 = =35/144 answer
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A box contains 5 white and 7 black balls, two successive draws of ball are made without replacements, what is the probability that the first ball is white and 2nd is black ball? First white : 5C1 / 12C1
2nd black : 7C1/ 11C1 =5/12 * 7/11 = =35/121 answer
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A problem in probability was given to 3 CS students : A,B,C their chances of solving are 1/3, 1/5 and ½ respectively, what is the probability that the question will be solved?
For these questions, start with opposites A will not solve : 2/3 B will not solve : 4/5 c will not solve : ½ none of them will solve 2/3 * 4/5 * ½ = 4/15 problem will be solved : 1-4/15 = 11/15
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From a group of students 30%, 40%, 50% failed in A/c, Eco, and at least one of these two papers. If an examinee is selected at random, what is the probability that he passed in A/c if it known that he failed in Eco.
50 failed in one of the two papers. 40% failed in eco and 30% in a/c. 30+40 = 70, which is 20 more than 50, so 20% failed in both the papers. 20 (40-20) failed in eco only, but not in a/c, so the required probability : 20/40 answer
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Out of 2 pots, first contains 3 red and 5 black balls, 2nd contains 4 red and 6 black. A ball is taken at random from 1st pot and is transferred to 2nd pot and now another ball is taken from 2nd pot, what is the probability the 2nd ball is red?
Now there are (4+6+1) =11 balls, but we dont know about 11th ball, which has been transferred from 1st pot. So there are two th possibilities for 11 ball: red / black (3/8*5c1/11c1)+(5/8*4c1/11c1) =35/88 answer
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by : DR. T.K. JAIN AFTERSCHO☺OL centre for social entrepreneurship sivakamu veterinary hospital road bikaner 334001 rajasthan, india for PGPSE participants PGPSE : FREE FOR ALL, OPEN FOR ALL mobile : 91+9414430763
5 DECEMBER 09 AFTERSCHOOOL centre for social entrepreneurship bikaner 1
WHAT IS EXCLUSIVE EVENT?
WHEN RESULTS FROM SOME EXPERIMENT ARE SUCH THAT IF ONE OUTCOME (EVENT) HAPPENS, OTHERS ARE IMPOSSIBLE, THEN THEY ARE CALLED EXCLUSIVE EVENTS. FOR EXAMPLE, IF YOU TOSS A COIN, YOU CAN GET EITHER HEAD OR TAIL – NOT BOTH - THESE ARE EXCLUSIVE EVENTS
5 DECEMBER 09 AFTERSCHOOOL centre for social entrepreneurship bikaner 2
WHAT IS PROBABILITY ?
FINDING THE POSSIBILTY OF AN EVENT OUT OF MANY EQUALLY LIKELY EVENTS IS CALLED PROBABILITY THUS IF YOU THROW A COIN, THE PROBABILITY OF GETTING HEAD IS ½ BECAUSE THERE ARE 2 EQUALLY LIKELY EVENTS AND ONE HEAD.
5 DECEMBER 09 AFTERSCHOOOL centre for social entrepreneurship bikaner 3
What is exhaustive list of events?
All the possibilities are called exhaustive list. For example : coin has two possibilities : head and tail – thus these two are exhaustive - there is no other possibilitiy
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AFTERSCHOOOL centre for social entrepreneurship bikaner
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Q : what is the probability of getting 2 heads out of 3 tosses of coins?
First : list down all the possibilities : H= Head, T=tail we may have all heads : HHH, we may have all tail : TTT : 1. HHH 2 TTT 3 HTH 4 THT 5 HHT 6 HTT 7 TTH 8 THH in simple terms : 2^3 = 8 possibilities : (the formula is options ^ number of trials) out of these the cases of 2 heads are : 3/8 answer
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What is the possibility of getting a difference of 2 when you toss a dice twice?
Let us list down all the possibilities : (1,1) (1,2)(13)(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), (6,1),(6,2),(6,3),(6,4),(6,5),(6,6) there are total 36 possibilities out of which, 8 cases involve difference of 2, so answer : 8/36
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What is the chance of picking up a spade or an ace from a pack of cards?
There are 13 spade cards. There are 4 ace cards. Thus there are 13+4 cards = 17 out of which one ace+spade is included thus we have 16 out of 52 possibilities : 16/52 answer
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AFTERSCHOOOL centre for social entrepreneurship bikaner
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What is the probability that a committee of 7 persons has 5 men and 2 women, it is to be formed out of 8 men and 5 ladies
We have to pick up 7 persons out of (8+5) = 13 members so the total possibilities are : 13C7 = 13! / (7! * (13-7)!) =13*12*11*10*9*8*7*6*5*4*3*2*1/ (7*6*5*4*3*2*1*6*5*4*3*2*1) =1716 methods of selecting 5 men and 2 women : 8C5 * 5C2 = (8*7*6/3*2*1) (5*4/2*1) = 560 answer = 560/1716 answer
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A number is picked up out of 25 natural numbers, what is the probability that it is divisible by 4?
Out of 25 number, there are 6 numbers divisible by 4 : 4,8,12,16,20,24 thus the probability is : 6/25 answer
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P (A-B) = 1/5 and P(A)=1/3 and P(B)=1/2, what is the probability that out of A and B, only B will happen?
P(A-B) = P(A) (1/3)– (P(A intersection B) = 1/5 thus P(A intersection B)= 1/3-1/5= 2/15 we want P (B-A) = P(B)-P(A intersection B) =1/2 – 2/15 = 11/30 ans.
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A hits 5 out of 9 correct shots and B hits 6 out of 11 correct, what is the probability that when they try they are able to hit correct?
First find out the possibility that target is NOT hit at all : A not hitting : (1-5/9) = 4/9 A = 4/9 and B = 5/11 A *B = 4/9 * 5/11 = 20/99 thus they will hit : 1-20/99 = 79/99 answer
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8 balls are distributed at random among 3 boxes, what is the probability that the first box will get 3 balls?
8 balls can be distributed to 3 boxes in 3^8 manners. We want first box to have 3 balls, so 8C3, now we have 5 balls left out which have to be distributed to remaining 2 boxes : 2^5 answer : (8C3*2^5)/3^8 = 1792/6561 answer
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Odds in favour of an event is 2:3 and odds against another event is 3:7 ,what is the probability that at least one of these will happen
First event : happening : 2/5 not happening : 3/5, 2nd event happen : 7/10 and not happen : 3/10 Find the probability that none of these happen : 3/5 * 3/10 = 9/50 at least one of them will happen is : 1-9/5 =41/50 answer
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Two balls are picked up from a bag containing 5 red and 7 blue balls at random, what is the probability that the balls are of 2 different colours?
1 out of 5 = 5C1 and 1 out of 7 = 7C1 total : 12C2 solution : (5*7)/66 =35/66 answer
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AFTERSCHOOOL centre for social entrepreneurship bikaner
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What is the probability of getting at least 7 in a single cast with 2 dice?
Let us list down all the possibilities : (1,1) (1,2)(13)(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), (6,1),(6,2),(6,3),(6,4),(6,5),(6,6) out of these : 16 cases have 7 or more than 7 as total so answer is : 21/36 or 7/12 answer
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What is the probabilty of having at least one 6 from 3 throws of a dice?
Having 0 six : 5/6*5/6*5/6* =125/216 having at least 1 six = 95/216
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If 2 letters are taken at random from the word home, what is the probabilty that none of the letters would be vowels?
Vowels are 2 and constants are 2. so : select 2 out of 2 divided by 2 out of 4. 2C2 / 4C2 2C2 = 1 and 4C2 = 6 so answer : 1/6 answer
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A,B,C are 3 mutually independent with probabilities .3, .2 and .4 respectively, what is P(AintersectionBintersectionC) ?
Aintersection B intersection C = .3*.2*.4 = .024 answer
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AFTERSCHOOOL centre for social entrepreneurship bikaner
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There are 10 balls numbered from 1 to 10 in a box if one of them is selected at random, what is the probability that the number printed on the ball would be an odd number greater than 4?
We have 5,7,9 as three numbers which are less than 10 but are more than 4 and are odd numbers. So the probability is 3/10 answer
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AFTERSCHOOOL centre for social entrepreneurship bikaner
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The wages of 8 workers in $ : 50,62,40,70,45,56,32,45, one worker is selected randomly, what is the probability that his wages would be lower than averge wages?
Total of all : 400, so divided by 8 gives 50. thus we have 4 values less than 50 so the answer is : 4/8 or 50% answer
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AFTERSCHOOOL centre for social entrepreneurship bikaner
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If an unbiased coin is tossed 3 times, what is the probability of getting more than 1 head?
List down all the possibilities : TOTAL NUMBER : 2^3 = 8 HHT,HHH,HTH,THH,TTT,TTH,THT,HTT = 4/8 or ½ answer
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If two dice are rolled, what is the probability of getting neither 6 nor 9 as total ?
Let us list down all the possibilities : (1,1) (1,2)(13)(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), (6,1),(6,2),(6,3),(6,4),(6,5),(6,6) total of 6 : 5 cases, total of 9 : 4 cases remaining cases : 27 probability : 27/36 = .75 answer
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What is the probability that 4 children selected at random have different birthdates ?
There are 365 days in a year. So the probabilities are : for the first date – we have 365 options, for 2nd date, we have 364 days (one date is reduced), for 3rd person, we have 343 days and so on. 365/365*364/365*363/365*362/365
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A box contains 5 white and 7 black balls, two successive draws of ball are made with replacements, what is the probability n d that the first ball is white and 2 is black ball?
First white : 5C1 / 12C1 2nd black : 7C1/ 12C1 =5/12 * 7/12 = =35/144 answer
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A box contains 5 white and 7 black balls, two successive draws of ball are made without replacements, what is the probability that the first ball is white and 2nd is black ball? First white : 5C1 / 12C1
2nd black : 7C1/ 11C1 =5/12 * 7/11 = =35/121 answer
5 DECEMBER 09 AFTERSCHOOOL centre for social entrepreneurship bikaner 25
A problem in probability was given to 3 CS students : A,B,C their chances of solving are 1/3, 1/5 and ½ respectively, what is the probability that the question will be solved?
For these questions, start with opposites A will not solve : 2/3 B will not solve : 4/5 c will not solve : ½ none of them will solve 2/3 * 4/5 * ½ = 4/15 problem will be solved : 1-4/15 = 11/15
5 DECEMBER 09 AFTERSCHOOOL centre for social entrepreneurship bikaner 26
From a group of students 30%, 40%, 50% failed in A/c, Eco, and at least one of these two papers. If an examinee is selected at random, what is the probability that he passed in A/c if it known that he failed in Eco.
50 failed in one of the two papers. 40% failed in eco and 30% in a/c. 30+40 = 70, which is 20 more than 50, so 20% failed in both the papers. 20 (40-20) failed in eco only, but not in a/c, so the required probability : 20/40 answer
5 DECEMBER 09 AFTERSCHOOOL centre for social entrepreneurship bikaner 27
Out of 2 pots, first contains 3 red and 5 black balls, 2nd contains 4 red and 6 black. A ball is taken at random from 1st pot and is transferred to 2nd pot and now another ball is taken from 2nd pot, what is the probability the 2nd ball is red?
Now there are (4+6+1) =11 balls, but we dont know about 11th ball, which has been transferred from 1st pot. So there are two th possibilities for 11 ball: red / black (3/8*5c1/11c1)+(5/8*4c1/11c1) =35/88 answer
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THANKS....
GIVE YOUR SUGGESTIONS AND JOIN AFTERSCHOOOL NETWORK / START AFTERSCHOOOL NETWORK IN YOUR CITY [email protected] PGPSE – WORLD'S MOST COMPREHENSIVE PROGRAMME IN SOCIAL ENTREPRENEURSHIP
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