Probability — Mathematics textbook solutions

9.1.1 Basic Terminologies

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Worked Example

Worked · 1

9.1.1a SolvedEx.1 $S$

Solved Examples

Worked · 3

9.1.1 SolvedEx.1 $Describe the sample space of the experiment when a coin and a die are thrown simultaneously.$
9.1.1 SolvedEx.2 $Sunita and Samrudhi who live in Mumbai wish to go on holiday to Delhi together. They can travel to Delhi from Mumbai either by car or by train or plane and on reaching Delhi they can go for city-tour either by bus or Taxi. Describe the sample space, showing all the combined outcomes of different ways they could complete city-tour from Mumbai.$
9.1.1 SolvedEx.3 $E_1$

9.1.5 Elementary Properties of Probability

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Solved Examples

Worked · 4

9.1.5 SolvedEx.1 $A \cup B \cup C = S$
9.1.5 SolvedEx.2 $One card is drawn at random from a pack of 52 cards. What is the probability that it is a King or Queen?$
9.1.5 SolvedEx.3 $Five employees in a company of 20 are graduates. If 3 are selected out of 20 at random. What is the probability that i) they are all graduates? ii) there is at least one graduate among them?$
9.1.5 SolvedEx.4 $The letters of the word STORY be arranged randomly. Find the probability that a) T and Y are together. b) arrangment begins with T and end with Y.$

Exercise 9.1

50 q

There are four pens: Red, Green, Blue and Purple in a desk drawer of which two pens are selected at random one after the other with replacement. State the sample space and the following events.
Ex 9.1 Q1(i) $A : Selecting at least one red pen.$
Ex 9.1 Q1(ii) $B : Two pens of the same color are not selected.$
A coin and a die are tossed simultaneously. Enumerate the sample space and the following events.
Ex 9.1 Q2(i) $A : Getting a Tail and an Odd number$
Ex 9.1 Q2(ii) $B : Getting a prime number$
Ex 9.1 Q2(iii) $C : Getting a head and a perfect square.$
Find $n(S)$ for each of the following random experiments.
Ex 9.1 Q3(i) $From an urn containing 5 gold and 3 silver coins, 3 coins are drawn at random$
Ex 9.1 Q3(ii) $5 letters are to be placed into 5 envelopes such that no envelop is empty.$
Ex 9.1 Q3(iii) $6 books of different subjects arranged on a shelf.$
Ex 9.1 Q3(iv) $3 tickets are drawn from a box containing 20 lottery tickets.$
Two fair dice are thrown. State the sample space and write the favorable outcomes for the following events.
Ex 9.1 Q4(i) $A : Sum of numbers on two dice is divisible by 3 or 4.$
Ex 9.1 Q4(ii) $B : Sum of numbers on two dice is 7.$
Ex 9.1 Q4(iii) $C : Odd number on the first die.$
Ex 9.1 Q4(iv) $D : Even number on the first die.$
Ex 9.1 Q4(v) $Check whether events A and B are mutually exclusive and exhaustive.$
Ex 9.1 Q4(vi) $Check whether events C and D are mutually exclusive and exhaustive.$
A bag contains four cards marked as 5, 6, 7 and 8. Find the sample space if two cards are drawn at random
Ex 9.1 Q5(i) $with replacement$
Ex 9.1 Q5(ii) $without replacement$
A fair die is thrown two times. Find the probability that
Ex 9.1 Q6(i) $sum of the numbers on them is 5$
Ex 9.1 Q6(ii) $sum of the numbers on them is at least 8$
Ex 9.1 Q6(iii) $first throw gives a multiple of 2 and second throw gives a multiple of 3.$
Ex 9.1 Q6(iv) $product of numbers on them is 12.$
Two cards are drawn from a pack of 52 cards. Find the probability that
Ex 9.1 Q7(i) $one is a face card and the other is an ace card$
Ex 9.1 Q7(ii) $one is club and the other is a diamond$
Ex 9.1 Q7(iii) $both are from the same suit.$
Ex 9.1 Q7(iv) $both are red cards$
Ex 9.1 Q7(v) $one is a heart card and the other is a non heart card$
Three cards are drawn from a pack of 52 cards. Find the chance that
Ex 9.1 Q8(i) $two are queen cards and one is an ace card$
Ex 9.1 Q8(ii) $at least one is a diamond card$
Ex 9.1 Q8(iii) $all are from the same suit$
Ex 9.1 Q8(iv) $they are a king, a queen and a jack$
From a bag containing 10 red, 4 blue and 6 black balls, a ball is drawn at random. Find the probability of drawing
Ex 9.1 Q9(i) $a red ball.$
Ex 9.1 Q9(ii) $a blue or black ball.$
Ex 9.1 Q9(iii) $not a black ball.$
A box contains 75 tickets numbered 1 to 75. A ticket is drawn at random from the box. Find the probability that,
Ex 9.1 Q10(i) $Number on the ticket is divisible by 6$
Ex 9.1 Q10(ii) $Number on the ticket is a perfect square$
Ex 9.1 Q10(iii) $Number on the ticket is prime$
Ex 9.1 Q10(iv) $Number on the ticket is divisible by 3 and 5$
Ex 9.1 Q11 $What is the chance that a leap year, selected at random, will contain 53 sundays?.$
Find the probability of getting both red balls, when from a bag containing 5 red and 4 black balls, two balls are drawn,
Ex 9.1 Q12(i) $with replacement$
Ex 9.1 Q12(ii) $without replacement$
A room has three sockets for lamps. From a collection 10 bulbs of which 6 are defective. At night a person selects 3 bulbs, at random and puts them in sockets. What is the probability that
Ex 9.1 Q13(i) $room is still dark$
Ex 9.1 Q13(ii) $the room is lit$
Letters of the word MOTHER are arranged at random. Find the probability that in the arrangement
Ex 9.1 Q14(i) $vowels are always together$
Ex 9.1 Q14(ii) $vowels are never together$
Ex 9.1 Q14(iii) $O is at the begining and end with T$
Ex 9.1 Q14(iv) $starting with a vowel and end with a consonant$
Ex 9.1 Q15 $4 letters are to be posted in 4 post boxes. If any number of letters can be posted in any of the 4 post boxes, what is the probability that each box contains only one letter?$
Ex 9.1 Q16 $15 professors have been invited for a round table conference by Vice chancellor of a university. What is the probability that two particular professors occupy the seats on either side of the Vice Chancellor during the conference.$
A bag contains 7 black and 4 red balls. If 3 balls are drawn at random find the probability that
Ex 9.1 Q17(i) $all are black$
Ex 9.1 Q17(ii) $one is black and two are red$

9.2 Addition Theorem

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Solved Examples

Worked · 2

9.2 SolvedEx.1 $Two dice are thrown together. What is the probability that, i) Sum of the numbers is divisible by 3 or 4? ii) Sum of the numbers is neither divisible by 3 nor 5?$
9.2 SolvedEx.2 $\frac{2}{3}$

Exercise 9.2

18 q

Ex 9.2 Q1 $First 6 faced die which is numbered 1 through 6 is thrown then a 5 faced die which is numbered 1 through 5 is thrown. What is the probability that sum of the numbers on the upper faces of the dice is divisible by 2 or 3?$
A card is drawn from a pack of 52 cards. What is the probability that,
Ex 9.2 Q2(i) $card is either red or black?$
Ex 9.2 Q2(ii) $card is either black or a face card?$
A girl is preparing for National Level Entrance exam and State Level Entrance exam for professional courses. The chances of her cracking National Level exam is 0.42 and that of State Level exam is 0.54. The probability that she clears both the exams is 0.11. Find the probability that
Ex 9.2 Q3(i) $She cracks at least one of the two exams.$
Ex 9.2 Q3(ii) $She cracks only one of the two.$
Ex 9.2 Q3(iii) $She cracks none.$
A bag contains 75 tickets numbered from 1 to 75. One ticket is drawn at random. Find the probability that,
Ex 9.2 Q4(a) $number on the ticket is a perfect square or divisible by 4$
Ex 9.2 Q4(b) $number on the ticket is a prime number or greater than 40$
Ex 9.2 Q5 $The probability that a student will pass in French is 0.64, will pass in Sociology is 0.45 and will pass in both is 0.40. What is the probability that the student will pass in at least one of the two subjects?$
Ex 9.2 Q6 $Two fair dice are thrown. Find the probability that number on the upper face of the first die is 3 or sum of the numbers on their upper faces is 6.$
For two events A and B of a sample space S, if $P(A) = \frac{3}{8}$ , $P(B) = \frac{1}{2}$ and $P(A \cup B) = \frac{5}{8}$ . Find the value of the following.
Ex 9.2 Q7(a) $P(A \cap B)$
Ex 9.2 Q7(b) $P(A' \cap B')$
Ex 9.2 Q7(c) $P(A' \cup B')$
Ex 9.2 Q8 $P(A \cup B) = \frac{5}{6}$
Ex 9.2 Q9 $m$
Form a group of 4 men, 4 women and 3 children, 4 persons are selected at random. Find the probability that,
Ex 9.2 Q10(i) $no child is selected$
Ex 9.2 Q10(ii) $exactly 2 men are selected$
Ex 9.2 Q11 $A number is drawn at random from the numbers 1 to 50. Find the probability that it is divisible by 2 or 3 or 10.$

9.3.1 Conditional Probability

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Solved Examples

Worked · 2

9.3.1 SolvedEx.1 $A card is drawn from a pack of 52 cards, given that it is a red card, what is the probability that it is a face card.$
9.3.1 SolvedEx.2 $A pair of dice is thrown. If sum of the numbers is an odd number, what is the probability that sum is divisible by 3?$

9.3.2 Conditional Probability of an Event

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Solved Examples

Worked · 2

9.3.2 SolvedEx.1 $Find the probability that a single toss of a die will result in a number less than 4 if it is given that the toss resulted in an odd number.$
9.3.2 SolvedEx.2 $P(A') = 0.7$

9.3.3 Multiplication Theorem

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Solved Examples

Worked · 2

9.3.3 SolvedEx.1 $Two cards are drawn from a pack of 52 cards one after other without replacement. What is the probability that both cards are ace cards?$
9.3.3 SolvedEx.2 $An urn contains 4 black and 6 white balls. Two balls are drawn one after the other without replacement, what is the probability that both balls are black?$

9.3.4 Independent Events

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Solved Examples

Worked · 8

Two cards are drawn at random one after the other. Given that first card drawn in non-face red card, what is the probability that second card is face card, if the cards are drawn
9.3.4 SolvedEx.1(i) $without replacement?$
9.3.4 SolvedEx.1(ii) $with replacement?$
If A and B are two independent events and $P(A) = \frac{3}{5}$ , $P(B) = \frac{2}{3}$ , find
9.3.4 SolvedEx.2(i) $P(A \cap B)$
9.3.4 SolvedEx.2(ii) $P(A \cap B')$
9.3.4 SolvedEx.2(iii) $P(A' \cap B)$
9.3.4 SolvedEx.2(iv) $P(A' \cap B')$
9.3.4 SolvedEx.2(v) $P(A \cup B)$
9.3.4 SolvedEx.3 $\frac{2}{9}$

Exercise 9.3

31 q

Ex 9.3 Q1 $A bag contains 3 red marbles and 4 blue marbles. Two marbles are drawn at random without replacement. If the first marble drawn is red, what is the probability the second marble is blue?$
Ex 9.3 Q2 $A box contains 5 green pencils and 7 yellow pencils. Two pencils are chosen at random from the box without replacement. What is the probability that both are yellow?$
Ex 9.3 Q3 $In a sample of 40 vehicles, 18 are red, 6 are trucks, of which 2 are red. Suppose that a randomly selected vehicle is red. What is the probability it is a truck?$
From a pack of well-shuffled cards, two cards are drawn at random. Find the probability that both the cards are diamonds when
Ex 9.3 Q4(i) $first card drawn is kept aside$
Ex 9.3 Q4(ii) $the first card drawn is replaced in the pack.$
A, B, and C try to hit a target simultaneously but independently. Their respective probabilities of hitting the target are $\frac{3}{4}$ , $\frac{1}{2}$ and $\frac{5}{8}$ . Find the probability that the target
Ex 9.3 Q5(a) $is hit exactly by one of them$
Ex 9.3 Q5(b) $is not hit by any one of them$
Ex 9.3 Q5(c) $is hit$
Ex 9.3 Q5(d) $is exactly hit by two of them$
The probability that a student X solves a problem in dynamics is $\frac{2}{5}$ and the probability that student Y solves the same problem is $\frac{1}{4}$ . What is the probability that
Ex 9.3 Q6(i) $the problem is not solved$
Ex 9.3 Q6(ii) $the problem is solved$
Ex 9.3 Q6(iii) $the problem is solved exactly by one of them$
Ex 9.3 Q7 $A speaks truth in 80% of the cases and B speaks truth in 60% of the cases. Find the probability that they contradict each other in narrating an incident.$
Two hundred patients who had either Eye surgery or Throat surgery were asked whether they were satisfied or unsatisfied regarding the result of their surgery. The following table summarizes their response.
Surgery Satisfied Unsatisfied Total
Throat 70 25 95
Eye 90 15 105
Total 160 40 200
If one person from the 200 patients is selected at random, determine the probability
Ex 9.3 Q8(a) $that the person was satisfied given that the person had Throat surgery$
Ex 9.3 Q8(b) $that person was unsatisfied given that the person had eye surgery$
Ex 9.3 Q8(c) $the person had Throat surgery given that the person was unsatisfied$
Ex 9.3 Q9 $Two dice are thrown together. Let A be the event 'getting 6 on the first die' and B be the event 'getting 2 on the second die'. Are the events A and B independent?$
The probability that a man who is 45 years old will be alive till he becomes 70 is $\frac{5}{12}$ . The probability that his wife who is 40 years old will be alive till she becomes 65 is $\frac{3}{8}$ . What is the probability that, 25 years hence,
Ex 9.3 Q10(a) $the couple will be alive$
Ex 9.3 Q10(b) $exactly one of them will be alive$
Ex 9.3 Q10(c) $none of them will be alive$
Ex 9.3 Q10(d) $at least one of them will be alive$
A box contains 10 red balls and 15 green balls. Two balls are drawn in succession without replacement. What is the probability that,
Ex 9.3 Q11(a) $the first is red and the second is green?$
Ex 9.3 Q11(b) $one is red and the other is green?$
A bag contains 3 yellow and 5 brown balls. Another bag contains 4 yellow and 6 brown balls. If one ball is drawn from each bag, what is the probability that,
Ex 9.3 Q12(a) $both the balls are of the same color?$
Ex 9.3 Q12(b) $the balls are of different color?$
Ex 9.3 Q13 $An urn contains 4 black, 5 white and 6 red balls. Two balls are drawn one after the other without replacement. What is the probability that at least one of them is black?$
Ex 9.3 Q14 $Three fair coins are tossed. What is the probability of getting three heads given that at least two coins show heads?$
Ex 9.3 Q15 $Two cards are drawn one after the other from a pack of 52 cards without replacement. What is the probability that both the cards drawn are face cards?$
Ex 9.3 Q16 $Bag A contains 3 red and 2 white balls and bag B contains 2 red and 5 white balls. A bag is selected at random, a ball is drawn and put into the other bag, and then a ball is drawn from that bag. Find the probability that both the balls drawn are of same color.$
Ex 9.3 Q17 $(Activity) : A bag contains 3 red and 5 white balls. Two balls are drawn at random one after the other without replacement. Find the probability that both the balls are white.$
Ex 9.3 Q18 $A family has two children. Find the probability that both the children are girls, given that at least one of them is a girl.$

Surgery	Satisfied	Unsatisfied	Total
Throat	70	25	95
Eye	90	15	105
Total	160	40	200

9.4 Bayes' Theorem

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Solved Examples

Worked · 3

9.4 SolvedEx.1 $A bag contains 6 red, 5 blue balls and another bag contains 5 red and 8 blue balls. A ball is drawn from first bag and without noticing colour is put in the second bag. A ball is drawn from the second bag. Find the probability that ball drawn is blue in colour.$
9.4 SolvedEx.2 $The chances of X, Y, Z becoming managers of a certain company are 4:2:3. The probabilities that the bonus scheme will be introduced if X, Y, Z become managers are 0.3, 0.5 and 0.8 respectively. If the bonus scheme has been introduced, what is the probability that X is appointed as the manager?$
9.4 SolvedEx.3 $The members of the consulting firm hire cars from three rental agencies, 60% from agency X, 30% from agency Y and 10% from agency Z. 9% of the cars from agency X need repairs, 20% of the cars from agency Y need repairs and 6% of the cars from agency Z need repairs. If a rental car delivered to the consulting firms needs repairs, what is the probability that it came from rental agency Y?$

Exercise 9.4

13 q

Ex 9.4 Q1 $There are three bags, each containing 100 marbles. Bag 1 has 75 red and 25 blue marbles. Bag 2 has 60 red and 40 blue marbles and Bag 3 has 45 red and 55 blue marbles. One of the bags is chosen at random and a marble is picked from the chosen bag. What is the probability that the chosen marble is red?$
Ex 9.4 Q2 $A box contains 2 blue and 3 pink balls and another box contains 4 blue and 5 pink balls. One ball is drawn at random from one of the two boxes and it is found to be pink. Find the probability that it was drawn from (i) first box (ii) second box.$
Ex 9.4 Q3 $There is a working women's hostel in a town, where 75% are from neighbouring town. The rest all are from the same town. 48% of women who hail from the same town are graduates and 83% of the women who have come from the neighboring town are also graduates. Find the probability that a woman selected at random is a graduate from the same town.$
Ex 9.4 Q4 $E_1$
Ex 9.4 Q5 $Jar I contains 5 white and 7 black balls. Jar II contains 3 white and 12 black balls. A fair coin is flipped; if it is Head, a ball is drawn from Jar I, and if it is Tail, a ball is drawn from Jar II. Suppose that this experiment is done and a white ball was drawn. What is the probability that this ball was in fact taken from Jar II?$
A diagnostic test has a probability 0.95 of giving a positive result when applied to a person suffering from a certain disease, and a probability 0.10 of giving a (false) positive result when applied to a non-sufferer. It is estimated that 0.5% of the population are sufferers. Suppose that the test is now administered to a person about whom we have no relevant information relating to the disease (apart from the fact that he/she comes from this population). Calculate the probability that:
Ex 9.4 Q6(a) $given a positive result, the person is a sufferer.$
Ex 9.4 Q6(b) $given a negative result, the person is a non-sufferer.$
Ex 9.4 Q7 $A doctor is called to see a sick child. The doctor has prior information that 80% of the sick children in that area have the flu, while the other 20% are sick with measles. Assume that there is no other disease in that area. A well-known symptom of measles is rash. From the past records, it is known that, chances of having rashes given that sick child is suffering from measles is 0.95. However occasionally children with flu also develop rash, whose chance are 0.08. Upon examining the child, the doctor finds a rash. What is the probability that child is suffering from measles?$
Ex 9.4 Q8 $\frac{9}{10}$
A box contains three coins: two fair coins and one fake two-headed coin is picked randomly from the box and tossed.
Ex 9.4 Q9(a) $What is the probability that it lands head up?$
Ex 9.4 Q9(b) $If happens to be head, what is the probability that it is the two-headed coin?$
Ex 9.4 Q10 $\frac{2}{5}$
Ex 9.4 Q11 $\frac{2}{7}$

9.5 Odds

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Solved Examples

Worked · 2

9.5 SolvedEx.1 $A fair die is thrown. What are the odds in favour of getting a number which is a perfect square in uppermost face of die?$
9.5 SolvedEx.2 $The probability of one event A happening is the square of the probability of second event B, but the odds against the event A are the cube of the odds against the event B. Find the probability of each event.$

Exercise 9.5

10 q

If odds in favour of X solving a problem are $4 : 3$ and odds against Y solving the same problem are $2 : 3$ . Find probability of:
Ex 9.5 Q1(i) $X solving the problem$
Ex 9.5 Q1(ii) $Y solving the problem$
Ex 9.5 Q2 $The odds against John solving a problem are 4 to 3 and the odds in favor of Rafi solving the same problem are 7 to 5. What is the chance that the problem is solved when both them try it?$
The odds against student X solving a statistics problem are $8 : 6$ and odds in favour of student y solving the same problem are $14 : 16$ . Find is the chance that
Ex 9.5 Q3(i) $the problem will be solved if they try it independently$
Ex 9.5 Q3(ii) $neither of them solves the problem$
The odds against a husband who is 60 years old, living till he is 85 are $7 : 5$ . The odds against his wife who is now 56, living till she is 81 are $5 : 3$ . Find the probability that
Ex 9.5 Q4(a) $at least one of them will be alive 25 years hence$
Ex 9.5 Q4(b) $exactly one of them will be alive 25 years hence.$
Ex 9.5 Q5 $7 : 4$
Ex 9.5 Q6 $In a single toss of a fair die, what are the odds against the event that number 3 or 4 turns up?$
Ex 9.5 Q7 $3 : 2$

Miscellaneous Exercise 9

40 q

(I) Select the correct option

Practice · 10

Select the correct answer from the given four alternatives.
Misc I Q1 $There are 5 girls and 2 boys, then the probability that no two boys are sitting together for a photograph is$
1. A. $\frac{1}{21}$
2. B. $\frac{4}{7}$
3. C. $\frac{2}{7}$
4. D. $\frac{5}{7}$
Select the correct answer from the given four alternatives.
Misc I Q2 $In a jar there are 5 black marbles and 3 green marbles. Two marbles are picked randomly one after the other without replacement. What is the possibility that both the marbles are black?$
1. A. $\frac{5}{14}$
2. B. $\frac{5}{8}$
3. C. $\frac{5}{8}$
4. D. $\frac{5}{16}$
Select the correct answer from the given four alternatives.
Misc I Q3 $Two dice are thrown simultaneously. Then the probability of getting two numbers whose product is even is$
1. A. $\frac{3}{4}$
2. B. $\frac{1}{4}$
3. C. $\frac{5}{7}$
4. D. $\frac{1}{2}$
Select the correct answer from the given four alternatives.
Misc I Q4 $In a set of 30 shirts, 17 are white and rest are black. 4 white and 5 black shirts are tagged as 'PARTY WEAR'. If a shirt is chosen at random from this set, the possibility of choosing a black shirt or a 'PARTY WEAR' shirt is$
1. A. $\frac{11}{15}$
2. B. $\frac{13}{30}$
3. C. $\frac{9}{13}$
4. D. $\frac{17}{30}$
Select the correct answer from the given four alternatives.
Misc I Q5 $There are 2 shelves. One shelf has 5 Physics and 3 Biology books and the other has 4 Physics and 2 Biology books. The probability of drawing a Physics book is$
1. A. $\frac{9}{14}$
2. B. $\frac{31}{48}$
3. C. $\frac{9}{38}$
4. D. $\frac{1}{2}$
Select the correct answer from the given four alternatives.
Misc I Q6 $\frac{2}{5}$
1. A. $\frac{34}{35}$
2. B. $\frac{1}{35}$
3. C. $\frac{8}{35}$
4. D. $\frac{27}{35}$
Select the correct answer from the given four alternatives.
Misc I Q7 $\frac{2}{3}$
1. A. $\frac{5}{6}$
2. B. $\frac{6}{7}$
3. C. $\frac{7}{8}$
4. D. $\frac{8}{9}$
Select the correct answer from the given four alternatives.
Misc I Q8 $Bag I contains 3 red and 4 black balls while another Bag II contains 5 red and 6 black balls. One ball is drawn at random from one of the bags and it is found to be red. The probability that it was drawn from Bag II.$
1. A. $\frac{33}{68}$
2. B. $\frac{35}{69}$
3. C. $\frac{34}{67}$
4. D. $\frac{35}{68}$
Select the correct answer from the given four alternatives.
Misc I Q9 $A fair is tossed twice. What are the odds in favour of getting 4, 5 or 6 on the first toss and 1, 2, 3 or 4 on the second die?$
1. A. $1 : 3$
2. B. $3 : 1$
3. C. $1 : 2$
4. D. $2 : 1$
Select the correct answer from the given four alternatives.
Misc I Q10 $The odds against an event are 5:3 and the odds in favour of another independent event are 7:5. The probability that at least one of the two events will occur is$
1. A. $\frac{52}{96}$
2. B. $\frac{71}{96}$
3. C. $\frac{69}{96}$
4. D. $\frac{13}{96}$

(II)

Practice · 30

The letters of the word 'EQUATION' are arranged in a row. Find the probability that
Misc II Q1(a) $All the vowels are together.$
Misc II Q1(b) $Arrangement starts wiht a vowel and ends wiith a consonant.$
Misc II Q2 $There are 6 positive and 8 negative numbers. Four numbers are chosen at random, without replacement, and multiplied. Find the probability that the product is a positive numbers.$
Misc II Q3 $Ten cards numbered 1 to 10 are placed in a box, mixed up thoroughly and then one card is drawn randomly. If it is known that the number on the drawn card is more than 3, what is the probability that it is and even number?$
Misc II Q4 $P(A \cap B) = \frac{1}{2}$
Misc II Q5 $If the letters of the word 'REGULATIONS' be arranged at random, what is the probability that there will be exactly 4 letters between R and E?$
In how many ways can the letters of the word ARRANGEMENTS be arranged?
Misc II Q6(a) $Find the chance that an arrangement chosen at random begins with the letters EE.$
Misc II Q6(b) $Find the probability that the consonants are together.$
Misc II Q7 $A letter is taken at random from the letters of the word 'ASSISTANT' and another letter is taken at random from the letters of the word 'STATISTICS'. Find probability that the selected letters are the same.$
Misc II Q8 $j$
Misc II Q9 $An urn contains 5 red balls and 2 green balls. A ball is drawn. If it's green a red ball is added to the urn and if it's red a green ball is added to the urn. (The original ball is not returned to the urn). Then a second ball is drawn. What is the probability the second ball is red?$
Misc II Q10 $The odds against A solving a certain problem are 4 to 3 and the odds in favor of solving the same problem are 7 to 5 find the probability that the problem will be solved.$
If $P(A) = P\left(\frac{A}{B}\right) = \frac{1}{5}$ , $P\left(\frac{B}{A}\right) = \frac{1}{3}$ then find
Misc II Q11(i) $P\left(\frac{A'}{B}\right)$
Misc II Q11(ii) $P\left(\frac{B'}{A'}\right)$
Let A and B be independent events with $P(A) = \frac{1}{4}$ , and $P(A \cup B) = 2P(B) - P(A)$ . Find
Misc II Q12(a) $P(B)$
Misc II Q12(b) $P(A/B)$
Misc II Q12(c) $P(B'/A)$
Misc II Q13 $Find the probability that a year selected will have 53 Wednesdays.$
The chances of P, Q and R, getting selected as principal of a college are $\frac{2}{5}, \frac{2}{5}, \frac{1}{5}$ respectively. Their chances of introducing IT in the college are $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}$ respectively. Find the probability that
Misc II Q14(a) $IT is introduced in the college after one of them is selected as a principal .$
Misc II Q14(b) $IT is introduced by Q.$
Misc II Q15 $Suppose that five good fuses and two defective ones have been mixed up. To find the defective fuses, we test them one-by-one, at random and without replacement. What is the probability that we are lucky and fine both of the defective fuses in the first two tests?$
Misc II Q16 $P(A \cup C) = \frac{2}{3}$
Misc II Q17 $The ratio of Boys to Girls in a college is 3:2 and 3 girls out of 500 and 2 boys out of 50 of that college are good singers. A good singer is chosen what is the probability that the chosen singer is a girl?$
Misc II Q18 $A and B throw a die alternatively till one of them gets a 3 and wins the game. Find the respective probabilities of winning. (Assuming A begins the game).$
Misc II Q19 $Consider independent trails consisting of rolling a pair of fair dice, over and over What is the probability that a sum of 5 appears before sum of 7?$
Misc II Q20 $A machine produces parts that are either good (90%), slightly defective (2%), or obviously defective (8%). Produced parts get passed through an automatic inspection machine, which is able to detect any part that is obviously defective and discard it. What is the quality of the parts that make it throught the inspection machine and get shipped?$
Misc II Q21 $Given three identical boxes, I, II and III, each containing two coins. In box I, both coins are gold coins, in box II, both are silver coins and in the box III, there is one gold and one silver soin. A person chooses a box at random and takes out a coin. If the coin is of gold, what is the probability that the other coin in the box is also of gold?$
Misc II Q22 $In a factory which manufactures bulbs, machines A, B and C manufacture respectively 25%, 35% and 40% of the bulbs. Of their outputs, 5, 4 and 2 percent are respectively defective bulbs. A bulbs is drawn at random from the product and is found to be defective. What is the probability that it is manufactured by the machine B?$
A family has two children. One of them is chosen at random and found that the child is a girl. Find the probability that
Misc II Q23(a) $both the children are girls.$
Misc II Q23(b) $both the children are girls given that at least one of them is a girl.$