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?
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
Find the probability that at least one head appears in Mathematics, 20 passed in Statistics and 10 passed in both. One student is selected at random. Find the two tosses of a coin. probability that he has passed, (I) at least in one of the subjects, (ii) none of the subjects, (iii) only in 3. A ball is drawn at random from a box containing 6 red Mathematics, (iv) only in Mathematics or only in balls, 5 white balls and 3 blue balls. Find the probability 2.
that it is (i) red; (ii) white; (iii) blue, (iv) not red; (v) red or white.
4.
5.
Statistics
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A box contains 8 red and 3 white balls. If 3 balls are 21. A die is thrown twice. Find the probability that the score drawn at random, find the probability that (i) all the in the first throw is 6 and the score in the second throw three are red; (ii) all three are white; (iii) 2 are red and is 5 or 6. 1 is white. 22. 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. 23.
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.
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 6. If four coins are tossed, find the probability of the second toss. occurrence of 2 heads and 2 tails. 7.
8.
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.
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 A committee of 4 boys and 3 girls is to be formed by 52 cards. Find the probability that they are both aces if lots from 8 boys and 5 girls. One of the boys is brother the first card is (i) replaced, (ii) not replaced. of one of the girls. Find the probability that both are included in the committee. 26. One bag contains 4 white and 2 black balls, another
contains 3 white 5 black balls. If one ball is drawn from Three groups of children contain (i) 3 girls and 1 boy; each bag. Find the probability that (i) both are white, (ii) 3 girls and 3 boys; (iii) 2 boys and 1 girl. One child (ii) both are black is selected at random from each group. Find the probability that the three selected children comprise of 27. A box contains 8 tickets bearing the numbers 1, 2, 3, 5, 1 girl and 2 boys. 7, 8, 9, 10. One ticket is drawn at random and kept aside, then the second is drawn. Find the probability 10. Five coins are tossed simultaneously. Find the that the both the tickets show odd numbers. probability that all of them do not give heads. 9.
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.
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?
12. Ninety balls are numbered from 1 to 90. Five balls are 29. A box contains 10 radio valves of which 4 are taken at random. What is the probability that there is at defective. Find the probability that if two valves are least one ball with a one-digit number? taken from the box, they are both defective. 13. Two dice are thrown. Find the probability that the sum 30. A purse contains 2 silver coins and 4 copper coins and of the numbers on the uppermost faces is either even another contains 4 silver coins and 3 copper coins. If a or perfect square. coin is selected at random from one of the two purses what the probability that it is a silver coin? 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 31. The probability that A can solve a problem is 4/5, that B ball drawn is red, blue or black. 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. 15. Not scored. 32. The odds that a book will be favourably reviewed by 16. One ticket is drawn at random from a lot of 20 tickets three independent critics are 3 to 2, 4 to 3 and 2 to 3 numbered from 1 to 20. What is the probability that the respectively. What is the probability that of the three number on the ticket drawn is divisible by (I) 2 or 3 reviews majority will be favourable? (ii) 3 or 7 33. The odds against A solving a problem are 8 to 6 and 17. A card is drawn from an ordinary deck of 52 wellodds in favour of B solving the same problem are 14 to shuffled cards. Find the probability that it is (i) neither red nor an ace; (ii) neither the ten of clubs nor 16. If both of them try the problem, what is the probability that (i) the problem will be solved, (ii) both A an ace. and B will solve the problem? 18. In a class of 100 students, 60 drink tea, 50 drink coffee 34. The Probability that a man will be alive after 25 years is and 30 drink both tea and coffee. A student from this 3/5 and the probability that his wife will be alive after 25 class is selected at random. What is the probability that years is 2/3. Find the probability that (i) both will be the student takes (i) at least one of the two drinks, alive, (ii) only the man will be alive, (iii) only the wife (ii) only one of the drinks? 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
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?
and a second prize. What is the probability that a man 49. A box contains 3 red, 4 green, 2 black and 1 white possessing 5 tickets wins a prize? 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 38. A man draws from an urn containing two balls, one white one Rs. 10. What is his mathematical white and one black. If he draws a white ball he wins. If expectation? 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 50. A die is tossed twice. If it shows the same number is repeated. What are his respective chances of twice, Mr. A gets Rs. 100, otherwise he loses Rs. 5. winning at 2nd, 3rd, 4th and 10th try. What is the mathematical expectation of Mr. A? 39. An urn contains four white and five black balls, a 51. A wheel of fortune at an amusement park is divided second urn contains five white and four black balls. into five colours, red, blue, green, yellow, brown. The One ball is transferred from the first to the second urn, probabilities of the spinner landing in any of these then a ball is drawn from the second urn. What the colors are 3/10, 3/10, 2/10, 1/10, 1/10 respectively. A probability that it is white? 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 40. In the example No. 39 suppose that two balls are to try her luck. What is her mathematical expectation? transferred from the first to the second urn. Find the probability that a ball then drawn from the second urn will be white. 52. Three envelopes are placed on a table. One contains a Rs. 10 note, the second a Rs. 20 note and the third 41. Not scored. 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. 42. Not scored. Otherwise he pays B the amount equal to the amount in the envelope. What is his mathematical expectation 43. Three units of A, B, C of a factory produce 25%, 25% if he guesses that the middle envelope contains Rs. 50 and 50"% of its production respectively. If the note? percentages of defective items produced by the three
units A, B, C are respectively 1%, 2% and 3% and an 53. A bag contains 10 coins. Of these 4 are 25 p. coins, 5 item selected at random is found to be defective, find are 20 p. coins and one X p. coin. A person draws out the probability that it is produced by the unit (i) A; (ii) B. a coin from this bag and his mathematical expectation is given to be 25 p. Find the value of X. 44. In a class of 100 students there are 60 boys and 40 girls, 20 boys and 10 girls failed in Mathematics. A roll 54. If a man purchases a raffle ticket he can win a first number selected at random is found to be that of a prize of Rs. 5000 or a second prize of Rs. 2000 with student who has failed in Mathematics. What is the probabilities .001 and 0.003. What should be a fair probability that it is of a girl? price to pay for the ticket. 45. The chance that a female worker in a chemical factory 55. A bag contains 2 white balls and 3 black balls. Four will contract an occupational disease is 0.04 and the persons A, B, C, D in that order each draws one ball chance for a male worker is 0.06. Out of 1000 workers and does not replace it. The first to draw a white ball in a factory 200 are females. One worker is selected at receives Rs. 10. Determine their expectations. random and the worker is found to have contracted the disease. What is the probability that the worker is a female? 56. A coin is tossed until a head appears. What is the expected number of tosses? 46. Suppose we have two machines I & II that produce shoes. Machine I makes 60% of shoes. The remaining 57. In a business venture a man can make a profit of Rs. are made by machine II. 10% of the shoes made by 2, 000 with a probability of 0.4 or have a loss of Rs. machine I are defective and 20% of the shoes made by 1000 with a probability of 0.6. What is his expected machine II are defective. A shoe was selected at profit. 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