Questions on Probability

1. If three cards are drawn at random from a pack of 52 cards what is the probability that all the three will be kings?

2. Find the probability that at least one head appears in two tosses of a coin.

3. A ball is drawn at random from a box containing 6 red balls, 5 white balls and 3 blue balls. Find the probability that it is (i) red; (ii) white; (iii) blue, (iv) not red; (v) red or white.

4. A box contains 8 red and 3 white balls. If 3 balls are drawn at random, find the probability that (i) all the three are red; (ii) all three are white; (iii) 2 are red and 1 is white.

5. A bag contains 3 white balls, 4 black balls and 5 red balls. If 3 balls are drawn at random find the probability that (I) all 3 are red; (ii) all 3 are black' (iii) 2 are red and 1 is black; (iv) 1 of each colour.

6. If four coins are tossed, find the probability of the occurrence of 2 heads and 2 tails.

7. Five cards are drawn from a pack of 52 well shuffled cards. Find the probability that (i) 4 are aces; (ii) 4 are aces and 1 is a king; (ii) 3 are tens and 2 are jacks; (iv) at least one face card us obtained.

8. A committee of 4 boys and 3 girls is to be formed by lots from 8 boys and 5 girls. One of the boys is brother of one of the girls. Find the probability that both are included in the committee.

9. Three groups of children contain (i) 3 girls and 1 boy; (ii) 3 girls and 3 boys; (iii) 2 boys and 1 girl. One child is selected at random from each group. Find the probability that the three selected children comprise of 1 girl and 2 boys.

10. Five coins are tossed simultaneously. Find the probability that all of them do not give heads.

11. If out of 100 tosses of a coin, 56 were heads, find the probability that a tail appears in the next toss of the coin.

12. Ninety balls are numbered from 1 to 90. Five balls are taken at random. What is the probability that there is at least one ball with a one-digit number?

13. Two dice are thrown. Find the probability that the sum of the numbers on the uppermost faces is either even or perfect square.

14. A bag contains 6 red, 5 blue, 3 white and 4 black balls. A ball is drawn at random. Find the probability that the ball drawn is red, blue or black.

15. Not scored.

16. One ticket is drawn at random from a lot of 20 tickets numbered from 1 to 20. What is the probability that the number on the ticket drawn is divisible by (I) 2 or 3 (ii) 3 or 7

17. A card is drawn from an ordinary deck of 52 well-shuffled cards. Find the probability that it is (i) neither red nor an ace; (ii) neither the ten of clubs nor an ace.

18. In a class of 100 students, 60 drink tea, 50 drink coffee and 30 drink both tea and coffee. A student from this class is selected at random. What is the probability that the student takes (i) at least one of the two drinks, (ii) only one of the drinks?

19. A marble is drawn at random from a box containing 10 red, 30 white, 20 blue and 15 orange marbles. Find the probability that it is (I) red or orange, (ii) neither red nor blue, (iii) not blue (iv) white, (v) red, white or blue.

20. Out of 80 students in a class, 30 passed in Mathematics, 20 passed in Statistics and 10 passed in both. One student is selected at random. Find the probability that he has passed, (I) at least in one of the subjects, (ii) none of the subjects, (iii) only in Mathematics, (iv) only in Mathematics or only in

Statistics.21. A die is thrown twice. Find the probability that the score

in the first throw is 6 and the score in the second throw is 5 or 6.

22. Suppose two dice are cast. Find the probability that the number on the uppermost face of the first die is even and that of the second is odd.

23. A fair die is tossed twice. Find the probability of getting 4, 5 or 6 on the first toss and 1, 2, 3, or 4 on the second toss.

24. A card is drawn at random from a full pack of cards under the assumption that the card drawn is a spade. Find the probability that it is a face card i.e. King, Queen or jack.

25. Two cards are drawn are from a well shuffled deck of 52 cards. Find the probability that they are both aces if the first card is (i) replaced, (ii) not replaced.

26. One bag contains 4 white and 2 black balls, another contains 3 white 5 black balls. If one ball is drawn from each bag. Find the probability that (i) both are white, (ii) both are black

27. A box contains 8 tickets bearing the numbers 1, 2, 3, 5, 7, 8, 9, 10. One ticket is drawn at random and kept aside, then the second is drawn. Find the probability that the both the tickets show odd numbers.

28. P can hit a target 4 times in 5 shots, Q, 3 times in 4 shots and R, twice in 3 shots,. They fire simultaneously. What is the probability that at least 2 shots hit?

29. A box contains 10 radio valves of which 4 are defective. Find the probability that if two valves are taken from the box, they are both defective.

30. A purse contains 2 silver coins and 4 copper coins and another contains 4 silver coins and 3 copper coins. If a coin is selected at random from one of the two purses what the probability that it is a silver coin?

31. The probability that A can solve a problem is 4/5, that B can solve it is 2/3 and that C can solve it is 3/7. If all of them try independently, find the probability that the problem will be solved.

32. The odds that a book will be favourably reviewed by three independent critics are 3 to 2, 4 to 3 and 2 to 3 respectively. What is the probability that of the three reviews majority will be favourable?

33. The odds against A solving a problem are 8 to 6 and odds in favour of B solving the same problem are 14 to 16. If both of them try the problem, what is the probability that (i) the problem will be solved, (ii) both A and B will solve the problem?

34. The Probability that a man will be alive after 25 years is 3/5 and the probability that his wife will be alive after 25 years is 2/3. Find the probability that (i) both will be alive, (ii) only the man will be alive, (iii) only the wife will be alive, (iv) at least one will be alive.

35. A and B play games of chess. A wins 6, B wins 4 and 2 games are drawn. They, then agree to play 3 more games. Find the probability that (i) A wins all the 3 games, (ii) 2 of the 3 games end in a draw.

36. The probability that a 50-year old man will be alive at 60 is 0.83 and the probability that a 45-year old woman

will be alive at 55 is 0.87. What is the probability that a man who is 50 and wife who is 45 will be alive 10 years hence?

37. There are 50 tickets in a lottery in which there is a first and a second prize. What is the probability that a man possessing 5 tickets wins a prize?

38. A man draws from an urn containing two balls, one white and one black. If he draws a white ball he wins. If he fails to draw a white ball, the draw is replaced, another black ball is added and he draws again. If he fails to draw a white ball in the next draw, the process is repeated. What are his respective chances of winning at 2nd, 3rd, 4th and 10th try.

39. An urn contains four white and five black balls, a second urn contains five white and four black balls. One ball is transferred from the first to the second urn, then a ball is drawn from the second urn. What the probability that it is white?

40. In the example No. 39 suppose that two balls are transferred from the first to the second urn. Find the probability that a ball then drawn from the second urn will be white.

41. Not scored.

42. Not scored.

43. Three units of A, B, C of a factory produce 25%, 25% and 50"% of its production respectively. If the percentages of defective items produced by the three units A, B, C are respectively 1%, 2% and 3% and an item selected at random is found to be defective, find the probability that it is produced by the unit (i) A; (ii) B.

44. In a class of 100 students there are 60 boys and 40 girls, 20 boys and 10 girls failed in Mathematics. A roll number selected at random is found to be that of a student who has failed in Mathematics. What is the probability that it is of a girl?

45. The chance that a female worker in a chemical factory will contract an occupational disease is 0.04 and the chance for a male worker is 0.06. Out of 1000 workers in a factory 200 are females. One worker is selected at random and the worker is found to have contracted the disease. What is the probability that the worker is a female?

46. Suppose we have two machines I & II that produce shoes. Machine I makes 60% of shoes. The remaining are made by machine II. 10% of the shoes made by machine I are defective and 20% of the shoes made by machine II are defective. A shoe was selected at random and was found to be non-defective. What is the probability that it was manufactured by machine I?

47. Johnny washes supper dishes twice a week and his elder brother Frank does them 5 times a week. Johnny's two days are chosen at random each week. The probability that Johnny will break one or more dishes during a washing is 0.1. The probability that Frank will is 0.02. One evening as the dishes were being washed their father heard a dish crash, He said. 'Apparently this is Johnny's dish day'. What is the probability that he was right ?

48. The incidence of Myxoedema (underactive thyroid gland) among people admitted to hospitals is about 1 in 1500. Doctors often use the 'protein bound iodine test' to determine whether or not a person has myxoedema. When the test is used on people who have myxoedema, it shows the presence of the disease in 90% of those tested. In the remaining 10% the test yields a false negative result. When the test is given to people who do not have myxoedema it show the absence of the disease in 99% of those tested. In the remaining 1% the test yields a false positive result. If the test is used on a hospital patient chosen at random, and the result is positive indicating that the

patient has myxoedema), what is the probability that the patient really has myxoedema?

49. A box contains 3 red, 4 green, 2 black and 1 white marbles. A man is blind folded and asked to select a marble. If he selects a red marble he gets Rs. 3, for a green one he wins Rs.2, for a black one Rs. 7 and for a white one Rs. 10. What is his mathematical expectation?

50. A die is tossed twice. If it shows the same number twice, Mr. A gets Rs. 100, otherwise he loses Rs. 5. What is the mathematical expectation of Mr. A?

51. A wheel of fortune at an amusement park is divided into five colours, red, blue, green, yellow, brown. The probabilities of the spinner landing in any of these colors are 3/10, 3/10, 2/10, 1/10, 1/10 respectively. A player can win Rs. 5 if it stops on red, Rs. 3 if it stops on blue, Rs. 4 if it stops on green, and lose Rs. 2 if it stops on yellow and Re. 1 if it stops on brown. M wants to try her luck. What is her mathematical expectation?

52. Three envelopes are placed on a table. One contains a Rs. 10 note, the second a Rs. 20 note and the third Rs. 50 note. A and B decide to play the following game. A will guess what is in the middle envelope. If he guesses correctly he gets the content of the envelope. Otherwise he pays B the amount equal to the amount in the envelope. What is his mathematical expectation if he guesses that the middle envelope contains Rs. 50 note?

53. A bag contains 10 coins. Of these 4 are 25 p. coins, 5 are 20 p. coins and one X p. coin. A person draws out a coin from this bag and his mathematical expectation is given to be 25 p. Find the value of X.

54. If a man purchases a raffle ticket he can win a first prize of Rs. 5000 or a second prize of Rs. 2000 with probabilities .001 and 0.003. What should be a fair price to pay for the ticket.

55. A bag contains 2 white balls and 3 black balls. Four persons A, B, C, D in that order each draws one ball and does not replace it. The first to draw a white ball receives Rs. 10. Determine their expectations.

56. A coin is tossed until a head appears. What is the expected number of tosses?

57. In a business venture a man can make a profit of Rs. 2, 000 with a probability of 0.4 or have a loss of Rs. 1000 with a probability of 0.6. What is his expected profit.