Students and Teachers Forum

1. If the first quartile of the data is 30.625, find the value of a.

C.I.

20 – 30

30 – 40

40 – 50

50 – 60

60 – 70

Frequency

8

a

5

4

3

Here,         First quartile (Q1) = 30.625, so Q1 class = 30 – 40.                     Cumulative frequency table of given data is given .....

2. According to the data given, find the maximum marks obtained by below 75% students.

C.I.

0 – 5

5 – 10

10 – 15

15 – 20

20 – 25

f

6

4

7

5

8

   The maximum marks obtained by below 75% students = Q3.                Cumulative frequency table of given data is given by:

3. If  = a and ∑fx=b, find the number of terms (N).

We have given,                  = a and ∑fx=b,                 Number of terms (N) = ? We have,     .....

4. If ∑fx = 370 + 20a, ∑f  = 23 + a and  = 17, find the value of a.

Given         ∑fx = 370 + 20a, ∑f  = 23 + a and  = 17.         a = ? We have,  Mean () = ∑fx∑f = 370+20a23+a Or, 17 = 370+20a23+a Or, 391 + 17a = 370 + 20a Or, 21 = 3a or, a = .....

5. If  = 50 and ∑fx = 750, find the number of terms (N).

Given,              = 50 and ∑ fx = 750,              Number of terms (N)= ? We have,  = ∑fxN or, 50=750N ∴N=75050=15, which is required .....

6. The daily wages of 32 workers are given below:
 
Daily wages (Rs.) 30-40 40-50 50-60 60-70 70-80
No. of workers 3 4 12 8 5

Find the median.

Here,   Daily wages (Rs.) No. of workers (f) Cumulative frequency (cf) 30-40 3 3 40-50 4 7 50-60 12 19 60-70 8 27 70-80 5 32   N = .....

7. If the first quartile class = 20- 30 and if (N4 - c.f) =8 then find the first quartile.

Given,             First quartile class = 20- 30 and if (N4 - c.f) =8, We have given,  The first quartile of the given series is given by,          .....

8. If the first quartile class = 20- 30 and if (N4 - c.f) =8 then find the first quartile.

Given,          First quartile class = 20- 30 and if (N4 - c.f) =8. The first quartile of the given series is given by,           First quartile = L + if (N4 - .....

9. If the third quartile class = 40 – 50 and (if) (3N4- - c.f)= 4, then find the third quartile.

Given,         Third quartile class = 40 – 50 and if (3N4 - c.f)=4. The third quartile of the given series is given by,                  Q3 = L + if (3N4 - .....

10. If  = 15, N = 32 and ∑fx = 350+13p, find the value of p.

Given,          = 15, N = 32 and ∑fx = 350+13p,         p = ? We have,  = ΣfxN             or, 15 = 350+13p 32   or, 350+13p = 32×15   or, 13p .....

11. Find the third quartile class from the given graph.

In the given curve, sum of frequencies (N) = 20. We have, Third quartile class = (3N4 )th item                                   = (34 ⨉ 20)th item   .....

12. In a continuous series, the value of assumed mean is a and the sum of frequencies (f) is N. If the deviation is (d), then find the mean  of the data.

We have given,             In a continuous series, the value of assumed mean is a and the sum of frequencies (f) is N &the deviation is (d),             .....

13. From the given curve, find the median class.

In the given curve, sum of frequencies (N) = 30. We have, median class = (N2 )th item                        = (302)th item                  .....

14. If mean () = 12, ∑fx = 70 + 10a and the number of frequency (N) = 5 + a, find the value of a.

Given,            Mean () = 12, ∑fx = 70 + 10a and the number of frequency (N) = 5 + a,            a = ? We have,  Mean () = ∑fxN = 70+10a5+a Or, 12 = 70+10a5+a Or, .....

15. If  = 50 and ∑fx = 750, find the number of terms (N).

Given,          = 50,          ∑fx = 750,         Number of terms (N) = ? We have,  = ∑fxN or, 50=750N ∴ N=75050=15, which is required .....

16. Find the first quartile class from the given graph.

In the given curve, sum of frequencies (N) = 20. We have, First quartile class =( N4)th item                                = (20 4)th item       .....

17. If  =32, N = 15 and ∑fx = k, find the value of k.

Given,          =32, N = 15 and ∑fx = k,         k = ?. We have,         = Σfx               .....

18. Two fair dice are thrown at random. What is the probability that the faces turn up show
               i) a sum of 7 
              ii) same faces
             iii) a sum less than 5
             iv) number 6 in the first die.

Solution: The sample space when two dice are thrown is:  

1st dice/ .....

19. Find the probability of getting both time head when a coin is tossed twice.

When a coin is tossed two times, the probability of getting head in 1st toss P(H) = 12  And the probability of getting head on 2nd toss is P(H) = 12 ∴ The probability of getting both head when a coin is tossed is       .....

20. A dice is thrown 1200 times and the records of outcomes are given in the table.
 
Outcomes 1 2 3 4 5 6
Frequency 186 205 211 187 204 207

Calculate the empirical probability of 
       i. less than 4 
      ii. greater than 3
     iii. even numbers
     iv. less than or equal to 3. 

Here, Total no. of trails n(S) = 1200 i.    Let E1 be the event of getting faces less than 4.       Then            n(E1) = 186 + 205 + 211               .....

21. There are 3 blue, 2 green and 1 yellow marbles of the same shape and size in a bag. If two marbles are drawn randomly in succession without replacement from the bag, show the probabilities of all possible outcomes in a tree diagram.

Let, G, Y and B denote the events of getting a ball of green, a ball of yellow and a ball of blue respectively. Then, the tree diagram is as shown in .....

22. A bag contains 6 black and 4 white marbles. A marble is drawn at random from the bag and the marble is replaced then, a second marble is drawn. By using a tree diagram, find the probability of getting the marbles of :
i.    both of the same colours and 
ii.   both of different colours.
 

Let, W and B denote the events of getting a ball of white and a ball of black respectively. Then, the tree diagram is as shown in figure. Now, from the tree diagram,     We have, probability of getting same colours= P(BB) + P(WW) = .....

23. X can solve 80% of the problems given in a book and Y can solve 75%. What is the probability that at least one of them will solve a problem, selected at random from the book?

Here,        The probability of solving a problem by student X, P(X) = 80 % = 0.8        The probability of solving a problem by student Y, P (Y) = 75% = 0.75. Now,         The .....

24. Find the probability of 1, 3 or 5 when a dice is rolled.

Here,          Sample space(S) = {1, 2, 3, 4, 5, 6}                              ∴n(S) = 6. Now, probability of getting 1, 3 or 5 is given .....

25. A coin is tossed two times in succession. Calculate the probability that both are tails by drawing a tree diagram.

Let, H and T denote the events of getting a head and a tail respectively. Then, the tree diagram is as shown in figure. Then from the diagram, probability that both are tails = P(TT) = .....

26. Two children are born in a family. By drawing a tree-diagram, find the sample space and also find the probability of having no daughter.

Let, S = son and D = daughter. This is the case of independent event.          Probability of a son, P(S) = 12          Probability of a daughter, P(D) = 12. Tree-diagram: Sample space: .....

27. There are 3 white, 4 blue and 5 green marbles in a box. Two drawings one at a time are made at random without replacement. Using tree-diagram, find the probability that
a) both balls are different color
b) both are not blue
c) both are not green.

Let, W = white ball, B = blue ball and G = green ball Then,         n(W) = 3, n(B) = 4 and n(G) = 5 This is the case of dependent event. Tree-diagram:  From the tree diagram,  P(both are same color) = P(WW) or .....

28. A card is drawn at random from a pack of 52 cards and at the same time a marble is drawn at random from a bag containing 2 red marbles and 3 blue marbles. 
         i. Find the probability of getting a blue marble and a king. 
        ii. Also find the probability getting king or blue marble. 

For cards;           Total no. of cards n(S) = 52 .  Let E1 be the event of getting king, then n(E1) = 4. ∴ Probability of getting king cards P(E1) =  452             .....

29. A coin is tossed three times successively. By the help of three diagram, 
   i) Find probability of getting all heads. 
  ii) Find the probability of getting at least 2 heads. 
 iii) Find the probability of getting at most 2 heads. 

No. of head = n(H) = 1 No. of tail = n(T)= 1. Tree diagram: Outcomes:            Probability: (H, H, H )             P(H, H, H)   = 1/2 × 1/2 × 1/2 = 1/8   .....

30. A bag contains 1 red, 1 blue and 1 white ball, three balls are drawn one after another without replacement. 
      i) Draw a tree diagram to show the probabilities of all possible outcomes. 
     ii) Find the probability of getting one ball of each colour. 

Here,         No. of red ball     = 1        No. of blue ball   = 1        No. of white ball = 1. The Probability of getting one ball of each colour is: P(R, B, W) + .....

31. A person is known to hit the target 3 out of 4 shots whereas another person is known to hit the target 2 out of 3 shots. Find the probability of the targets being hit when they both try. Also find the probability that the target is not hit at all when they both try.

    Let A and B be the event that the 1st and 2nd person hit the target. Now,             P(A) = 34            P(A)ˈ = 1 - 34  = 14.           .....

32. A bag contains 5 blue balls, 3 yellow balls and 2 white balls. If two balls are drawn at random without replacement, calculate the probability that:
a) both are blue
b) both are same color.

Let, B = black pen, Y = yellow pen and W = white pen Then, n(B) = 5, n(Y) = 3 and n(W) = 2 Probability of an ace card, P(A) = 452  Probability of a non-face card, P(N) = 4852. Tree-diagram: From the diagram, a) Both are blue = .....

33. The probability that A can solve a problem is 1/5, what is the probability that A cannot solve the problem?

Given,        The probability that A can solve a problem is 15 . We know,        Probability that A can not solve = P(A') = 1 - P(A)                   .....

34. A bag contains one white, one blue and one green ball. A ball is drawn at random and is not replaced. Then another ball is drawn. Using tree-diagram, find the probability of getting a green ball either in the first draw or in the second draw.

Let, W = white ball, B = blue ball and G = green ball     Then,             n(W) = 1, n(B) = 1 and n(G) = 1 This is the case of dependent event. Tree-diagram: Sample space = {GB, GW, BG, BW, WG, .....

35. There are 6 black balls and 3 white balls in a box. Two balls are drawn randomly one after another (without replacement). Show the probabilities of the possible outcomes using tree-diagram. Also find the probabilities that
(a) both are white balls
(b) the first is black and the second is white.

Let, B = black balls and W = white balls. Then,        n(R) = 6 and n(W) = 3. Tree-diagram: This is the case of dependent event. a) From the tree diagram, P(both white) = P(WW)               .....

36. A bag contains 5 blue balls, 3 yellow balls and 2 white balls. If two balls are drawn at random without replacement, calculate the probability that:
a) both are blue
b) both are same color

Let, B = blue balls, Y = yellow balls and W = white balls. Then,          n(B) = 5, n(Y) = 3 and n(W) = 2. Probability of an ace card, P(A) =  452 Probability of a non-face card, P(N) = .....

37. Two cards are drawn from a well shuffled deck of 52 cards in succession. Find the probability that
i) both cards are aces
ii) both cards are not aces if the first card is not replaced before the second is drawn. (Use tree diagram)

Let A = ace cards and N = non-ace cards Then, n(A) = 4 and n(N) = 48 Tree-diagram: This is the case of dependent event. From the tree diagram, P(both aces) = P(AA) = 452 X 351                    .....

38. Six green and three blue balls of the identical size are kept in a box. Two draws are made one by one without replacement. Find the probability using a tree diagram that
a) both balls are green
b) both balls are blue
c) one is green and the other is blue.

Let, G = green  and B = blue Then, n(G) = 6 and n(B) = 3 This is the case of dependent event. Probability of a green ball, P(G) = 69 Probability of a blue ball, P(B) = 39 Tree-diagram: a) Sample space: {BB, BG, GG, GB}. b) Probability .....

39. A ball is drawn from a bag containing three red and two white balls randomly and replaced before the second is drawn. Then another ball is drawn. Find the probabilities that
a) both balls are red
b) first is white and second is red.

Solution: Let, R = red balls  and W = white balls. Then, n(R) = 3 and n(W) = 2. Tree-diagram: This is the case of independent event. Probability of a red ball, P(R) = 35 Probability of a white ball, P(W) = 25 From the tree .....

40. If three children are born in a family. Calculate the probability that all children are girls by drawing a tree diagram.

Let, G and B denote the events of giving birth to a girl and a boy respectively. Then, the tree diagram is as shown in figure. From the diagram, probability that all are girls= P(GGG) = .....

41. From a class having 24 boys and 15 girls, two students are selected randomly. Show all the probabilities of being a boy and a girl on a tree diagram.

Let, B and G denote the events of selecting a boy and a girl respectively. Then, the probabilities are shown in tree diagram. Possible outcomes = { GG, GB, BG, .....

42. A bag contains 8 blue and 4 red marbles. Two marbles are chosen in succession at random (without replacement) from the bag. Using tree-diagram
a) find the probability that both are blue marbles
b) find the probability that both are red marbles
c) find the probability that both marbles are of the same color.

Let , B = blue  and R = red Then, n(B) = 8 and n(R) = 4 This is the case of dependent event. Probability of a blue ball, P(B) = 812 Probability of a red ball, P(R) = 412 Tree-diagram: a) Probability that both balls are blue, P(BB) = P(B) .....

43. What is the probability of getting 3 on the dice and head on the coin when dice is rolled and a coin is tossed simultaneously?

Here, for a dice,           Sample space, S1 = {1, 2, 3, 4, 5, 6}. For a coin,           Sample space, S2 = {Head, tail}. Now, probability of getting 3 on the dice and head on the coin is .....

44. There are 3 white and 6 green balls of the same size in a box. Two draws each of one ball (without replacement) are made. Using tree-diagram, find the probability that
a) one is white and the other is green and
b) at least one of the balls is white.

Let ,W = white  and G = green Then, n(W) = 3 and n(G) = 6 This is the case of dependent event. Probability of a white ball, P(W) = 39  Probability of a green ball, P(G) = 69 Tree-diagram: a) Probability that one is white and the .....

45. A card is drawn from well shuffled pack of 52 cards.
What is the probability that the card so drawn is either a king or a black?
 

Let A be the event of king cards and B be the events of black cards.  Then,         Number of king cards n(A) = 4          Number of black cards n(B) = 26          Number of .....

46. A bag contains balls numbered 1 to 20, one ball is drawn at random. What is the probability that it is multiple of 4 and 5?

Let A and B be the set of balls numbered with multiple of 4 and 5 respectively.  Then,          S = {1, 2, ........., 20}         A = {4, 8, 12, 16, 20}          B = {5, 10, .....

47. A card is drawn randomly from a pack of cards numbered from 2 to 25. Find the probability of getting a card having the number exactly divisible by 4 or 7.

Let A and B be the set of cards divisible by 4 and 7 respectively.  Then,        S = {2, 3, 4, ..........., 25}         A = {4, 8, 12, 16, 20, 24}         B = {7, 14, .....

48. A bag contains 5 yellow, 3 blue and 2 green balls. If a ball is drawn randomly from the bag, find the probability of not getting a blue ball.

Given,         Number of yellow balls = 5          Number of blue balls = 3          Number of green balls = 2          Total number of balls = 5 + 3 + .....

49. From a packet of 52 cards. Two cards are drawn one after another without replacement. Use a tree diagram to find the probability of getting, 
       i) both face cards. 
      ii) both none face cards. 
     iii) one of each kind. 

Given,         No. of face cards = 12         No. of non face cards = 40         Total no. cards = 52.      Let E be the event of getting face cards and Eˈ be the event of .....

50. A number card is drawn randomly from the set of numbered cards, numbered from 6 to 39. Find the probability that the card may be a prime number or a cube number.

Let A and B are sets of prime number and cube number cards respectively.           ∴ S = {6, 7, 8, ............, 39}             A = {7, 11, 13, 17, 19, 23, 29, 31, 37}     .....