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# Probability Tricks & Tips

In everyday life, we come across the situations having either some certainty or uncertainty and we are generally interested to measure this certainty or uncertainty. We want to measure that up to what extent this particular situation would occur. We generally achieve it qualitatively; not able to calculate it quantitatively. So what about if we want to measure it quantitatively? That is achieved with the help of the Theory of Probability.

**Probability is the Measure of Uncertainty.**

*Experiment:*An operation which can produce some well-defined outcomes is called an experiment.

*Random Experiment:*

An experiment in which all possible outcomes are know and the exact output cannot be predicted in advance, is called a random experiment.

*Examples:*

- Rolling an unbiased dice.
- Tossing a fair coin.
- Drawing a card from a pack of well-shuffled cards.
- Picking up a ball of certain colour from a bag containing balls of different colours.

It has 13 cards of each suit, name *Spades, Clubs, Hearts and Diamonds*.

Cards of spades and clubs are *black cards*.

Cards of hearts and diamonds are *red cards*.

There are 4 honours of each unit.

There are *Kings, Queens and Jacks*. These are all called *face cards*.

*Sample Space:*

When we perform an experiment, then the set S of all possible outcomes is called the *sample space*

In tossing a coin, S = {H, T}

If two coins are tossed, the S = {HH, HT, TH, TT}.

In rolling a dice, we have, S = {1, 2, 3, 4, 5, 6}.

**Probability of Occurrence of an Event:**

Let** S** be the sample space and **E** be the event

Then

1) 𝑷(𝑺) = 𝟏

2) ≤ 𝟎𝑷(𝑬) ≤ 𝟏

3) 𝑷(𝑬) = 𝟏 → 𝑬 𝒊𝒔 𝒄𝒂𝒍𝒍𝒆𝒅 𝒂 𝒔𝒖𝒓𝒆 𝑬𝒗𝒆𝒏𝒕

4) 𝑷(𝑬) = 𝟎 → 𝑬 𝒊𝒔 𝒄𝒂𝒍𝒍𝒆𝒅 𝑰𝒎𝒑𝒐𝒔𝒔𝒊𝒃𝒍𝒆

5) 𝑷(𝑬) + 𝑷(𝑬̅) = 1

**EXAMPLE:-**A fair coin is tossed at random. Find the probability of getting:

1) Head

2) Tail

**EXAMPLE:-** Two unbiased coins are tossed simultaneously at random.

Find the probability of getting:

1) Head on the first coin.

2) Head on the second coin.

3) Head on both the coins.

4) No heads.

5) At least one head.

6) At most one head.

**Practice Set On Probability**

- A bag contains 6 red, 2 blue and 4 green balls. 3 balls are chosen at random. What is the probability that at least 2 balls chosen will be red?

A) 2/7

B) 1/2

C) 1/3

D) 2/5

E) 3/7 - Tickets numbered 1 to 250 are in a bag. What is the probability that the ticket drawn has a number which is a multiple of 4 or 7?

A) 83/250

B) 89/250

C) 77/250

D) 93/250

E) 103/250 - From a deck of 52 cards, 3 cards are chosen at random. What is the probability that all are face cards?

A) 14/1105

B) 19/1105

C) 23/1105

D) 11/1105

E) 26/1105 - One 5 letter word is to be formed taking all letters – S, A, P, T and E. What is the probability that this the word formed will contain all vowels together?

A) 2/5

B) 3/10

C) 7/12

D) 3/5

E) 5/12 - One 5-digit number is to be formed from numbers – 0, 1, 3, 5, and 6 (repetition not allowed). What is the probability that number formed will be even?

A) 8/15

B) 7/16

C) 7/15

D) 3/10

E) 13/21

**Directions (6-8):** There are 3 bags containing 3 colored balls – Red, Green and Yellow.

Bag 1 contains:

24 green balls. Red balls are 4 more than blue balls. Probability of selecting 1 red ball is 4/13

Bag 2 contains:

Total balls are 8 more than 7/13 of balls in bag 1. Probability of selecting 1 red ball is 1/3. The ratio of green balls to blue balls is 1 : 2

Bag 3 contains:

Red balls equal total number of green and blue balls in bag 2. Green balls equal total number of green and red balls in bag 2. Probability of selecting 1 blue ball is 3/14.

- 1 ball each is chosen from bag 1 and bag 2, What is the probability that 1 is red and other blue?

A) 15/128

B) 21/115

C) 17/135

D) 25/117

E) 16/109 - Some green balls are transferred from bag 1 to bag 3. Now probability of choosing a blue ball from bag 3 becomes 3/16. Find the number of remaining balls in bag 1.

A) 60

B) 58

C) 52

D) 48

E) 44 - Green balls in ratio 4 : 1 from bags 1 and 3 respectively are transferred to bag 4. Also 4 and 8 red balls from bags 1 and 3 respectively . Now probability of choosing green ball from bag 4 is 5/11. Find the number of green balls in bag 4?

A) 12

B) 15

C) 10

D) 9

E) 11

**Directions (9-10):** There are 3 people – A, B and C. Probability that A speaks truth is 3/10, probability that B speaks truth is 3/7 and probability that C speaks truth is 5/6. For a particular question asked, at most 2 people speak truth. All people answer to a particular question asked.

- What is the probability that B will speak truth for a particular question asked?

A) 7/18

B) 14/33

C) 4/15

D) 9/28

E) 10/33 - A speaks truth only when B does not speak truth, then what is the probability that C does not speak truth on a question?

A) 11/140

B) 21/180

C) 22/170

D) 13/140

E) None of these

- There are 100 tickets in a box numbered 1 to 100. 3 tickets are drawn at one by one. Find the probability that the sum of number on the tickets is odd.

A) 2/7

B) 1/2

C) 1/3

D) 2/5

E) 3/7 - There are 4 green and 5 red balls in first bag. And 3 green and 5 red balls in second bag. One ball is drawn from each bag. What is the probability that one ball will be green and other red?

A) 85/216

B) 34/75

C) 95/216

D) 35/72

E) 13/36 - A bag contains 2 red, 4 blue, 2 white and 4 black balls. 4 balls are drawn at random, find the probability that at least one ball is black.

A) 85/99

B) 81/93

C) 83/99

D) 82/93

E) 84/99 - Four persons are chosen at random from a group of 3 men, 3 women and 4 children. What is the probability that exactly 2 of them will be men?

A) 1/9

B) 3/10

C) 4/15

D) 1/10

E) 5/12 - Tickets numbered 1 to 120 are in a bag. What is the probability that the ticket drawn has a number which is a multiple of 3 or 5?

A) 8/15

B) 5/16

C) 7/15

D) 3/10

E) 13/21 - There are 2 people who are going to take part in race. The probability that the first one will win is 2/7 and that of other winning is 3/5. What is the probability that one of them will win?

A) 14/35

B) 21/35

C) 17/35

D) 19/35

E) 16/35 - Two cards are drawn at random from a pack of 52 cards. What is the probability that both the cards drawn are face card (Jack, Queen and King)?

A) 11/221

B) 14/121

C) 18/221

D) 15/121

E) 14/221 - A committee of 5 people is to be formed from among 4 girls and 5 boys. What is the probability that the committee will have less number of boys than girls?

A) 7/12

B) 7/15

C) 6/13

D) 5/12

E) 7/13 - A bucket contains 2 red balls, 4 blue balls, and 6 white balls. Two balls are drawn at random. What is the probability that they are not of same color?

A) 5/11

B) 14/33

C) 2/5

D) 6/11

E) 2/3 - A bag contains 5 blue balls, 4 black balls and 3 red balls. Six balls are drawn at random. What is the probability that there are equal numbers of balls of each color?

A) 11/77

B) 21/77

C) 22/79

D) 13/57

E) 15/77

**Directions (1-3): An urn contains some balls colored white, blue and green. The probability of choosing a white ball is 4/15 and the probability of choosing a green ball is 2/5. There are 10 blue balls.**

- What is the probability of choosing one blue ball?

A) 2/7

B) 1/4

C) 1/3

D) 2/5

E) 3/7 - What is the total number of balls in the urn?

A) 45

B) 34

C) 40

D) 30

E) 42 - If the balls are numbered 1, 2, …. up to number of balls in the urn, what is the probability of choosing a ball containing a multiple of 2 or 3?

A) 3/4

B) 4/5

C) 1/4

D) 1/3

E) 2/3 - There are 2 brothers A and B. Probability that A will pass in exam is 3/5 and that B will pass in exam is 5/8. What will be the probability that only one will pass in the exam?

A) 12/43

B) 19/40

C) 14/33

D) 21/40

E) 9/20 - If three dices are thrown simultaneously, what is the probability of having a same number on all dices?

A) 1/36

B) 5/36

C) 23/216

D) 1/108

E) 17/216 - There are 150 tickets in a box numbered 1 to 150. What is the probability of choosing a ticket which has a number a multiple of 3 or 7?

A) 52/125

B) 53/150

C) 17/50

D) 37/150

E) 32/75 - There are 55 tickets in a box numbered 1 to 55. What is the probability of choosing a ticket which has a prime number on it?

A) 3/55

B) 5/58

C) 8/21

D) 16/55

E) 4/13 - A bag contains 4 white and 5 blue balls. Another bag contains 5 white and 7 blue balls. What is the probability of choosing two balls such that one is white and the other is blue?

A) 61/110

B) 59/108

C) 45/134

D) 53/108

E) 57/110 - The odds against an event are 2 : 3 and the odds in favor of another independent event are 3 : 4. Find the probability that at least one of the two events will occur.

A) 11/35

B) 27/35

C) 13/35

D) 22/35

E) 18/35 - The odds against an event are 1 : 3 and the odds in favor of another independent event are 2 : 5. Find the probability that one of the event will occur.

A) 17/28

B) 5/14

C) 11/25

D) 9/14

E) 19/28

- From a pack of 52 cards, 1 card is chosen at random. What is the probability of the card being diamond or queen?

A) 2/7

B) 6/15

C) 4/13

D) 1/8

E) 17/52 - From a pack of 52 cards, 1 card is drawn at random. What is the probability of the card being red or ace?

A) 5/18

B) 7/13

C) 15/26

D) 9/13

E) 17/26 - There are 250 tickets in an urn numbered 1 to 250. One ticket is chosen at random. What is the probability of it being a number containing a multiple of 3 or 8?

A) 52/125

B) 53/250

C) 67/125

D) 101/250

E) 13/25 - There are 4 white balls, 5 blue balls and 3 green balls in a box. 2 balls are chosen at random. What is the probability of both balls being non-blue?

A) 23/66

B) 5/18

C) 8/21

D) 7/22

E) 1/3 - There are 4 white balls, 3 blue balls and 5 green balls in a box. 2 balls are chosen at random. What is the probability that first ball is green and second ball is white or green in color?

A) 1/3

B) 5/18

C) 1/2

D) 4/21

E) 11/18 - 2 dices are thrown. What is the probability that there is a total of 7 on the dices?

A) 1/3

B) 2/7

C) 1/6

D) 5/36

E) 7/36 - 2 dices are thrown. What is the probability that sum of numbers on the two dices is a multiple of 5?

A) 5/6

B) 5/36

C) 1/9

D) 1/6

E) 7/36 - There are 25 tickets in a box numbered 1 to 25. 2 tickets are drawn at random. What is the probability of the first ticket being a multiple of 5 and second ticket being a multiple of 3.

A) 5/11

B) 1/4

C) 2/11

D) 1/8

E) 3/14 - What is the probability of selecting a two digit number at random such that it is a multiple of 2 but not a multiple of 14?

A) 17/60

B) 11/27

C) 13/30

D) 31/60

E) 17/30 - There are 2 urns. 1st urn contains 6 white and 6 blue balls. 2nd urn contains 5 white and 7 black balls. One ball is taken at random from first urn and put to second urn without noticing its color. Now a ball is chosen at random from 2nd urn. What is the probability of the second ball being a white colored ball?

A) 11/13

B) 6/13

C) 5/13

D) 5/12

E) 11/12

**1. A bag contains 12 white and 18 black balls. Two balls are drawn in succession without replacement. What is the probability that first is white and second is black?**

**A)**36/135

**B)**36/145

**C)**18/ 91

**D)**30/91

**E)**None of these

**2. Two dice are thrown simultaneously. What is the probability of getting two numbers whose product is even?**

**A)**3/16

**B)**1/8

**C)**3/4

**D)**1/2

**E)**None of these

**3. In a class, there are 15 boys and 10 girls. Three students are selected at random. The probability that 1 girl and 2 boys are selected is:**

**A)**21/46

**B)**21/135

**C)**42/135

**D)**Can’t be determined

**E)**None of these

**4. A card is drawn from a pack of 52 cards. The probability of getting a queen of club or a king of heart is?**

**A)**3/26

**B)**3/52

**C)**1/26

**D)**1/4

**E)**None of these

**5. A bag contains 4 white, 5 red and 6 blue balls. Three balls are drawn at random from the bag. The probability that all of them are blue, is:**

**A)**1/91

**B)**2/91

**C)**3/91

**D)**4/91

**E)**None of these.

**6. A bag contains 2 yellow, 3 green and 2 blue balls. Two balls are drawn at random. What is the probability that none of the balls drawn is blue?**

**A)**5/7

**B)**1/21

**C)**10/21

**D)**2/9

**E)**None of these

**7. Three coins are tossed. What is the probability of getting at most two tails?**

**A)**1/8

**B)**5/8

**C)**3/8

**D)**7/8

**E)**None of these

**One card is drawn at random from a pack of 52 cards. What is the probability that the card drawn is a face card (Jack, Queen and King only)?**

**A)**1/13

**B)**2/13

**C)**3/13

**D)**3/52

**E)**None of these

**9. P and Q sit in a ring arrangement with 10 persons. What is the probability that P and Q will sit together?**

**A)**2/11

**B)**3//11

**C)**4/11

**D)**5/11

**E)**None of these

**10. Two dice are thrown simultaneously. Find the probability of getting a multiple of 2 on one dice and multiple of 3 on the other dice.**

**A)**1/9

**B)**11/36

**C)**13/36

**D)**Data inadequate

**E)**None of these

**Answers**

**1.**B

**2.**C

**3.**A

**4**. C

**5**. D

**6**. C

**7.**D

**8**. C

**9**. A

**10**.B

**Explanation:**

**1.**The probability that first ball is white= 12c1/30c1= 2/5

Since, the ball is not replaced; hence the number of balls left in bag is 29.

Hence the probability the second ball is black= 18c1/29c1 =18/29

Required probability = 2/5*18/29 = 36/145

**2.**In a simultaneous throw of two dice, we have n(S) = (6 x 6) = 36.

Then, E = {(1, 2), (1, 4), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 2), (3, 4),

(3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 2), (5, 4), (5, 6), (6, 1),

(6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}

n(E) = 27.

so probability = 27/36 = 3/4

**3.**Probability = 10c1*15c2/25c3

= 21/46

**4**. 2/52 = 1/26

**5.**6c3/15c3 =4/91

**6.** 5c2/7c2 = 10/21

**7.** 7/8

**8.** 12/52 =3/13

**9. **n(S)= number of ways of sitting 12 persons at round table:

=(12-1)!=11!

Since two persons will be always together, then number of persons:

=10+1=11

So, 11 persons will be seated in (11-1)!=10! ways at round table and 2 particular persons will be seated in 2! ways.

n(A)= The number of ways in which two persons always sit together =10!×2

So probability = 10!*2!/11!= 2/11

**10**. 11/36

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