Class 11 RD Sharma Solutions — Глава 33 Вероятность — Упражнение 33.3 | Набор 1

Вопрос 1. Какое из следующих утверждений не может быть допустимым присвоением вероятности элементарным событиям исходов выборочного пространства S={w ₁ , w ₂ , w ₃ , w ₄ , w ₅ , w ₆ , w ₇ } :

Элементарными событиями являются:

Решение:

We know that  the value of probability of an event must be between 0 and 1 and that the sum of all probabilities of a event must be equal to one. Using this rationale,

1. This is valid since each probability of event w₁ lies between 0 and 1 and the sum of all probabilities of event w₁ is 1, or P{w₁}= 1.

2. This is valid since each probability of event w₂ lies between 0 and 1 and the sum of all probabilities of event w₂  is 1, or P{w₂}=2.

3. This is not valid as the sum of all probabilities is 2.8, which is more than one. P{w_i} = 2.8

4. This is not valid since the probability of w₇ is 15/14= 1.07 > 1.

Hence, the valid events are only 1 and 2.

Вопрос 2. Брошена игральная кость. Найдите вероятность получить:

(i) простое число

Решение:

Since a die is thrown, the number of outcomes in sample space must be 6. Thus, n{S}= 6.

Let E be the event of getting a prime number.

∴ E = {2,3,5} or n{E} = 3

We know, Probability of an event = Number of outcomes in that event / number of outcomes in sample space

∴ P{E} = n{E}/ n{S} = 3/6

∴ P{E}= 1/2

(ii) 2 или 4

Решение:

Let N be the event of getting a 2 or 4.

Hence, N = (2,4} or n{N} = 2

Probability of event N or P{N} = n{N}/n{S}

or P{N} = 2/6

∴ P{N} = 1/3

(iii) кратное 2 или 3

Решение:

Let R be the event of getting a multiple of 2 or 3 when a die is thrown.

∴ R = {2,3,4,6}

Thus, n{R} = 4

∴ P{R} = n{R}/ n{S} = 4/6

∴ P{R} = 2/3

Вопрос 3: При одновременном броске пары игральных костей найти вероятность выпадения:

(i) 8 как сумма

Решение:

Since a pair of die have been thrown, the total number of outcomes in the sample space S become

6×6 = 6²= 36.

Let R be the event of getting 8 as the sum when a pair of dice is thrown together.

∴ R = {(2,6), (3,5), (4,9), (5,3), (6,2)}

∴ n{R} = 5

Hence probability of R = n{R}/ n{S}

∴ P{R} = 5/36

(ii) дублет

Решение:

Let D be the event that a doublet appear on the dice when a pair of dice is thrown together.

∴ D = { (1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}

Thus, n{D} = 6

Hence probability of D = n{D}/ n{S}

P{D} = 6/36 = 1/6

∴ P{D} = 1/6

(iii) дублет простых чисел

Решение:

Let F be the event of getting a doublet of prime numbers when a pair of dice is thrown together.

∴ F = {(2,2), (3,3), (5,5)}

Thus, n{F} = 3

Hence probability of F = n{F}/ n{S}

P{F}= 3/36 = 1/12

∴ P{F} = 1/12

(iv) дублет нечетных чисел

Решение:

Let Q be the event of getting a doublet of odd numbers when a pair of dice is thrown.

∴ Q = {(1,1), (3,3), (5,5)}

Thus, n{Q} = 3

 Probability of Q = n{Q}/ n{S}

P{Q} = 3/36 = 1/12

∴ P{Q} = 1/12

(v) сумма больше 9

Решение:

Let K be the event of getting a sum greater than 9 when a pair of dice is thrown.

∴ K = {(4,6), (5,5), (5,6), (6,4), (6,5), (6,6)}

Thus, n{K} = 6

Probability of K = n{K}/n{S}

P{K}= 6/36 = 1/6

∴ P{K} = 1/6

(vi) четное число на первом

Решение:

Let W be the event of getting an even number on first dice when a pair of dice is thrown. If the first number has to be even, it means that any other number can appear on the second dice, be it even or odd. Using this rationale:

∴ W ={(2,1), (2,2), (2,3), (2,4),(2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

 Thus, n{W} = 18

 Probability of W = n{W}/ n{S}

= 18/36 = 1/2

∴ P{W} = 1/2

(vii) четное число на одном и кратное 3 на другом

Решение:

Let V be the event of that we get an even number on one and a multiple of 3 on the other die when a pair of dice is thrown simultaneously. In this case, it is not specified that the even number shall have to be on the face of first or the second dice.

So long as an even number appears on any of the two dices, it is to be taken as an outcome of the given event. Same goes for multiple of 3 as well. Using this rationale:

∴ V = {(2,3), (2,6), (4,3), (4,6), (6,3), (6,6), (3,2), (3,4), (3,6), (6,2), (6,4)}

Thus, n{V} = 11

Probability of V = n{V}/ n{S}

∴ P{V} = 11/36

(viii) ни 9, ни 11 как сумма чисел на гранях

Решение:

Let H be the event of getting neither 9 nor 11 as the sum of the numbers on the faces of two dice when they are thrown together.

Therefore, H’ shall be the event that either 9 or 11 appear as sum of the two numbers on the faces of the two dices.

H’ = {(3,6), (4,5), (5,4), (5,6), (6,3), (6,5)}

n{H’} = 6

P{H’} = n{H’}/ n{S} = 6/36 = 1/6                  

Thus, P{H} = 1−P{H’} = 1−1/6 = 5/6                            

(ix) сумма менее 6

Решение:

Let E be the event of obtaining a sum less then 6 when a pair of die is thrown together.

∴ E = {(1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (3,1), (3,2), (4,1)}

∴ n{E} =10

Probability of E = P{E} = n{E}/ n{S} = 10/36 = 5/18

∴ P(E) = 5/18

(x) сумма меньше 7

Решение:

Let C be the event that a sum less than 7 appears on the faces of the two dice when thrown together.

∴ C = {(1,1), (1,2), (1,3), (1,4), (1,5), (2,1), (2,2),(2,3),(2,4),(2,5),(3,1),(3,2),(3,3).(4,1),(4,2),(5,1)}

Thus, n{C} = 15

Probability of C = P{C} = n{C}/ n{S}

= 15/36 = 5/12

∴ P(C) = 5/12

(xi) сумма больше 7

Решение:

Let X be an event of getting a sum more than 7 from the numbers on the faces of the two dice when thrown together.

∴ X = {(2,6), (3,5), (3,6), (4,4), (4,5), (4,6), (5,3),(5,4), (5,5), (5,6), (6,2), (6,3), (6,4), (6,5), (6,6)}

Thus n{X} = 15

Probability of X = P{X} = n{X}/ n{S} = 15/36 = 5/12

∴ P(X) = 5/12

(xii) ни дублет, ни всего 10

Решение:

Let A be the event of getting neither a doublet nor a total of 10 when a pair of dice is thrown simultaneously.

∴ A’ is the event of getting either a doublet or a total of 10 on the faces of the two dices.

∴ A’ = {(1,1), (2,2), (3,3), (4,6), (5,5), (6,4), (6,6)}

Thus, n{A’} = 8

Probability of A’ = n{A’}/ n{S} = 8/36 = 2/9

Probability of A = P{A} = 1−P{A’}

= 1−2/9

∴ P(A’} = 7/9

(xiii) нечетное число на первом и 6 на втором

Решение:

Let B be the event of getting an odd number on the first and 6 on the second face of the two dices.

∴ B = {(1,6), (3,6), (5,6)}

Thus n{B} = 3

Probability of B = P{B} = n{B}/ n{S} = 3/36 = 1/12

∴ P(B} = 1/12

(xiv) число больше 4 с каждой стороны

Решение:

Let J be the event of getting a number greater than 4 on each side of dice when both dices are thrown simultaneously.

∴ J = {(5,5), (5,6), (6,5), (6,6)}

Thus n{J} = 4

Probability of J = P{J} = n{J}/ n{S} = 4/36 = 1/9

∴ P(J} = 1/9

(xv) всего 9 или 11

Решение:

Let I be the event of getting a total of 9 or 11 when two dice are thrown.

∴ I = {(3,6),(4,5),(5,4),(5,6),(6,3),(6,5)}

Thus n{I} = 6

Probability of I = P{I} = n{I}/ n{S} = 6/36 = 1/6

∴ P(I} = 1/6                                  

(xvi) сумма больше 8

Решение:

Let Z be the event of getting a total greater than 8.

∴ Z = {(3,6),(4,5),(4,6),(5,4),(5,5),(5,6),(6,3),(6,4),(6,5),(6,6)}

∴ n{Z} = 10

Probability of Z = n{Z}/ n{S} = 10/36 = 5/18              

∴ P(Z} = 5/18                                                  

Вопрос 4. Найдите вероятность того, что при одном броске трех игральных костей выпадет 17 или 18 очков.

Решение:

Since three dice have been thrown simultaneously, the total number of outcomes in that sample space become 6×6×6= 6³ = 216.

Let E be the event of getting a total of 17 or 18. Thus E = {(6,6,5), (6,5,6), (5,6,6), (6,6,6)}

Hence, n{E} = 4

Probability of event E = P{E} = n{E}/ n{S} = 4/216 = 1/54

∴ P{E} = 1/54

Вопрос 5. Подбрасывают три монеты. Найдите вероятность получить:

(i) ровно 2 головы

Решение:

Since 2 coins have been tossed together, the total outcomes of the sample space are n{S} =

 2×2×2 = 2³ = 8.

Let A be the event of getting exactly 2 heads.

∴ A = {HHT, HTH, THH} or, n{A} = 3            

Probability of event A = P{A} = n{A}/n{S} = 3/8    

∴ P{A} = 3/8

(ii) не менее двух головок

Решение:

Let B be the event of getting at least 2 heads when three coins are thrown together.

∴ B = {HHH, HHT, THH, HTH} or, n{B} = 4

Probability of event B = P{B} = n{B}/ n{S} = 4/8 = 1/2

∴ P{B} = 1/2

(iii) по крайней мере, одна голова и один хвост

Решение:

Let C be the event of getting at least one head and one tail .

C= {(HHT, THT, HTT, TTH, HTH, THH} or, n{C}= 6

P{C}= n{C}/ N(S) = 6/8 = 3/4

∴ P{C}= 3/4      

Вопрос 6. Какова вероятность того, что в обычном году 53 воскресенья?

Решение:

We know that an ordinary year has 52 weeks and 1 day.

52 weeks in a year obviously implies 52 Sundays.

But in the question, we have to find the probability of having 53 Sundays which means that we have to fine the probability of the last 1 day of an ordinary year to be a Sunday.

Total number of days in a week = Number of outcomes of sample space = n(S) = 7

S= {MONDAY, TUESDAY, WEDNESDAY, THURSDAY, FRIDAY, SATURDAY, SUNDAY}

Hence, the probability of that one day being a Sunday = 1/7

Thus, the probability of having 53 Sundays in an ordinary year is 1/7.

Вопрос 7. Какова вероятность того, что в високосном году 53 воскресенья и 53 понедельника?

Решение:

We know that in a leap year we have 52 weeks and 2 days (366 days)

The sample space for the last 2 days shall be

S= {(Monday, Tuesday), (Tuesday, Wednesday), (Wednesday, Thursday), (Thursday, Friday), (Friday, Saturday), (Saturday, Sunday), (Sunday, Monday)} or, n{S}=7

Let E denote the event of getting a Sunday and a Monday as the last 2 days in a leap year.

∴ E= {Sunday, Monday} or, n{E} = 1

Probability of E= P{E}= n{E}/ n{S} = 1/7

Thus, the probability that a leap year has 53 Sundays and 53 Mondays is 1/7.

Вопрос 8. В мешке 8 красных и 5 белых шаров. Наугад извлекаются три шара. Найдите вероятность того, что:

а) все три шара белые

Решение:

Since there are 8+5= 13 balls in the bag and we are required to draw any three balls at random.

Total number of outcomes shall be= n{S}= ¹³C₃ = 286.

Let A be the event that all the three balls drawn are white.

Hence the number of outcomes in event A= n{A} = ⁵C₃= 10.

Probability of event A= n{A}/n{S} = 10/286= 5/143

Hence the probability of drawing all three white balls is 5/143.

б) все три шара красные

Решение:

Let R denote the event that all three balls drawn are red. 

Since total number of red balls is 8, the number of ways of drawing 3 red balls out of 8 = ⁸C₃ = 56 = n{R}.

We know, n{S}= 286

∴ Probability of R= n{R}/n{S}= 56/286 = 28/143

Hence, the probability of drawing all three red balls is 28/143.

в) один шар красный и два белых

Решение:

Let E be the event that one ball drawn is red and the other two balls are white.

Number of outcomes of E= ⁸C₁ × ⁵C₂ = 8 ×10 = 80.

∴ Probability of E= n{E}/n{S} = 80/286 = 40/143

Hence the probability of getting one red and other two balls as white is 40/143.

Вопрос 9. Найдите вероятность того, что при одном броске трех игральных костей выпадут одинаковые числа на всех трех костях.

Решение:

Since three dice have been rolled together, the total number of outcomes in the sample space= n{S}= 6³= 6×6×6= 216.

Let E be the event of getting the same numbers on all the three dice.

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

Number of outcomes in E= n{E} = 6

Probability of E= n{E}/n{S}= 6/216 = 1/36

Hence the probability of getting the same numbers on all the three dice is 1/36.

Вопрос 10. Бросают вместе две игральные кости. Найдите вероятность того, что сумма чисел на кубиках больше 10.

Решение:

Since two dice have been rolled together, the total number of outcomes in the sample space= n{S}= 62= 6×6= 36.

Let E be the event of getting a total of numbers on the dice greater than 10.

∴E= {(5,6), (6,5), (6,6)}

Number of outcomes in E= n{E}= 3

Since n{S}= 52, Probability of E= n{E}/n{S}= 3/36 = 1/12

Hence the probability of getting a total of numbers on the dice greater than 10  is 1/12.

Вопрос 11. Из колоды в 52 карты случайным образом вытягивается карта. Найдите вероятность того, что вынутая карта:

(i) черный король

Решение:

Since  a card is drawn from a pack of 52 cards, so number of elementary events in the sample space = n {S} = ⁵²C₁ = 52.

Let E be the event of drawing a black king. Since there are two black kings, one of spade and other of club,

∴n {E} = ²C₁ = 2

Since n{S}= 52, Probability of event E = n{E}/n{S} = 2/52 = 1/26.

Hence the probability of drawing a black king is 1/26.

(ii) либо черная карта, либо король

Решение:

Let A be the event of drawing a black card or a king. We know that there are 26 black cards and 4 kings out of which 2 kings are black.

Hence, total number of outcomes of event A= n{E} = ²⁶C₁ + ⁴C₁ – ²C₁= 28

In this case, 2 needs to be subtracted from total because there are two black kings which are already counted in black cards and to avoid double counting.

Since n{S}= 52, Probability of A= n{A}/n{S} = 28/52 = 7/13.

Hence, the probability of getting either a black card or a king is 7/13.

(iii) черные и король

Решение:

Let W be the event of drawing a black card and a king. We know there are two black kings, one of spade and other of club.

Hence number of outcomes of W= n{W} = ²C₁ = 2.

Since n{S}=52, Probability of W= n{W}/n{S} = 2/52 = 1/26.

Hence the probability of drawing a black card and a king is 1/26.

(iv) валет, дама или король

Решение:

Let B be the event of drawing a jack, queen or king. We know there are 4 kings, 4 queens and 4 jacks in a deck of cards.  

Hence total number of outcomes of event B = n{B} = ⁴C₁ + ⁴C₁ + ⁴C₁ = 12.

Since n{S}= 52, Probability of B = P{B} = n{B}/n{S} = 12/52 = 3/13

Hence the probability of drawing a jack, queen or king is 3/13.

(v) ни сердца, ни короля

Решение:

Let L be the event of drawing neither a heart nor a king and let L′ as the event that either a heart or king appears.

Since there are 13 hearts and 4 kings total number of outcomes = n (L′) = ⁶C₁ + ⁴C₁ – 1=16 ,deducting the king of hearts.

Probability of L’ = P{L′} = n{L’}/n{S} = 16/52 = 4/13

So, P{L} = 1 – P{L’} = 1 – 4/13 = 9/13

Hence the probability of drawing  neither a heart nor a king is 9/13.

(vi) пика или туз

Решение:

Let D be the event of drawing a spade or king. We know there are 13 spades and 4 kings, but 1 king is already included in the 4 kings.

Number of outcomes in D = n{D} = ¹³C₁ + ⁴C₁ – 1=16

Since N{S} = 52, Probability of event D = P{D} = n{D}/n{S} = 16/52 = 4/13

Hence the probability of drawing either a spade or an ace is 4/13.

(vii) ни туз, ни король

Решение:

Let K be the event of drawing neither an ace nor a king and K′ as the event that either an ace or king appears.

Number of outcomes in K’ = n{K’} = ⁴C₁ + ⁴C₁ = 8

Since n{S}= 52, Probability of K’ = P{K’} = n{K’}/n{S} = 8/52= 2/13

So, P{K} = 1 – P{K’} = 1 – 2/13= 11/13

Hence the probability of drawing neither an ace nor a king is 11/13.

(viii) алмазная карта

Решение:

Let X be the event of drawing a diamond card. We know there are 13 diamond cards.

Hence number of outcomes of event X = n{X} = ¹³C₁ = 13.

Since n{S} = 52, Probability of event X = P{X} = n{X}/n{S} = 13/52 = 1/4

Hence the probability of drawing a diamond card is 1/4.

(ix) не алмазная карта

Решение:

Let X be the event of drawing not a diamond card and X′ as the event that diamond card appears. We know there are 13 diamond cards.

Hence number of outcomes of event X = n{X} = ¹³C₁ = 13.

Since n{S} = 52, Probability of event X = P{X} = n{X}/n{S} = 13/52 = 1/4

So, P{X’} = 1 – P{X’} = 1 – 1/4 = 3/4

Hence the probability of not drawing a diamond card is 3/4.

(х) черная карта

Решение:

Let E be the event of drawing a black card. We know there are 26 black cards (spades and clubs).

Hence total number of outcomes of event E = n{E} = ²⁶C₁ = 26

Since n{S} = 52, Probability of event E = P{E} = n{E}/n{S} = 26/52= 1/2

Hence the probability of drawing a black card is 1/2.

(xi) не туз

Решение:

Let E be the event of drawing not an ace and  E′ as the event that ace card appears. We know that there are 4 ace cards.

Number of outcomes in E’ = n{E’} = ⁴C₁ = 4.

Since n{S} = 52, Probability of event E’ = P{E’} = n{E’}/n{S} = 4/52 = 1/13

So, P (E) = 1 – P (E′) = 1 – 1/13 =12/13

Hence the probability of not drawing an ace is 12/13.

(xii) не черная карта

Решение:

Let E be the event of not drawing a black card. We know there are 26 cards other than black cards (red cards of hearts and diamonds)

Hence number of outcomes of drawing a red card = n{E} = ²⁶C₁ = 26 .

Since n{S} = 52, Probability of event E = P{E} = n{E}/n{S} = 26/52 = 1/2.

Hence the probability of not drawing a black card is 1/2.

Вопрос 12. При перетасовке колоды из 52 игральных карт случайно выпадают четыре. Найдите вероятность того, что недостающие карты будут по одной каждой масти.

Решение:

A pack of 52 cards from which 4 are dropped. We now have to find the probability that the missing cards should be one from each suit.

We know that, from well shuffled pack of cards, 4 cards missed out total possible outcomes = n{S} = ⁵²C₄ = 270725

Let E be the event that four missing cards are from each suite

Hence total number of outcomes of event E = n{E} = ¹³C₁ × ¹³C₁ × ¹³C₁ × ¹³C₁ = 134

Probability of event E =P{E} = n{E}/n{S} = 134/270725 = 2197/20825

Hence the probability that the missing cards should be one from each suit is 2197/20825.

Вопрос 13. Из колоды в 52 карты берутся одновременно четыре карты. Найдите шанс, что это будут четыре награды одной масти.

Решение:

We have to find the probability that all the face cards  drawn from a pack of 52 cards are of same suits.

Total possible outcomes in sample space =n{S} = ⁵²C₄

Let E be the event that all the cards drawn are face cards of same suit.

Hence number of outcomes of event E = n (E)= 4 × ⁴C₄ = 4 × 1 = 4.

Probability of event E =P{E} = n{E}/n{S}= 4/270725

Hence the probability that the face cards  drawn from a pack of 52 cards are of same suits is 4/270725.

Вопрос 14. Билеты с номерами от 1 до 20 смешиваются между собой, после чего случайным образом вытягивается билет. Какова вероятность того, что номер билета кратен 3 или 7?

Решение:

We have to find the probability that the ticket has a number which is a multiple of 3 or 7.

Total possible outcomes in sample space =n{S} = ²⁰C₁ = 20.

Let E be the event of getting ticket which has number that is multiple of 3 or 7.

Hence outcomes of event E are {3,6,9,12,15,18,7,14}. Number of outcomes of event E are n{E} = 8.

Probability of event E =P{E} = n{E}/n{S}= 8/20 = 4/5

Therefore, the probability that the ticket has a number which is a multiple of 3 or 7 is 4/5.

Вопрос 15. В мешке 6 красных, 4 белых и 8 синих шаров. Если случайно вынуть три шара, найти вероятность того, что один из них красный, один белый и один синий.

Решение:

We have to find the probability that the one is red, one is white and one is blue.

Since three balls are drawn so, total number of outcomes for drawing 3 balls is n{S} = ¹⁸C₃ = 816

Let E be the event that one red, one white and one blue ball is drawn.

n{E} = ⁶C₁ × ⁴C₁ × ⁸C₁ = 192

Probability of event E =P{E} = n{E}/n{S} = 192 / 816 = 4/17

Therefore, the probability that one is red, one is white and one is blue is 4/17.