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CHAPTER 1 Basic Probability
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1.51. Work Problem 1.50 with no replacement after each drawing.
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Conditional probability and independent events
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1.52. A box contains 2 red and 3 blue marbles. Find the probability that if two marbles are drawn at random (without replacement), (a) both are blue, (b) both are red, (c) one is red and one is blue. 1.53. Find the probability of drawing 3 aces at random from a deck of 52 ordinary cards if the cards are (a) replaced, (b) not replaced. 1.54. If at least one child in a family with 2 children is a boy, what is the probability that both children are boys 1.55. Box I contains 3 red and 5 white balls, while Box II contains 4 red and 2 white balls. A ball is chosen at random from the first box and placed in the second box without observing its color. Then a ball is drawn from the second box. Find the probability that it is white.
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1.56. A box contains 3 blue and 2 red marbles while another box contains 2 blue and 5 red marbles. A marble drawn at random from one of the boxes turns out to be blue. What is the probability that it came from the first box 1.57. Each of three identical jewelry boxes has two drawers. In each drawer of the first box there is a gold watch. In each drawer of the second box there is a silver watch. In one drawer of the third box there is a gold watch while in the other there is a silver watch. If we select a box at random, open one of the drawers and find it to contain a silver watch, what is the probability that the other drawer has the gold watch 1.58. Urn I has 2 white and 3 black balls; Urn II, 4 white and 1 black; and Urn III, 3 white and 4 black. An urn is selected at random and a ball drawn at random is found to be white. Find the probability that Urn I was selected.
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1.59. A coin is tossed 3 times. Use a tree diagram to determine the various possibilities that can arise. 1.60. Three cards are drawn at random (without replacement) from an ordinary deck of 52 cards. Find the number of ways in which one can draw (a) a diamond and a club and a heart in succession, (b) two hearts and then a club or a spade. 1.61. In how many ways can 3 different coins be placed in 2 different purses
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1.62. Evaluate (a) 4 P2, (b) 7 P5, (c) 10 P3. 1.63. For what value of n is n 1 P3
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n P4
1.64. In how many ways can 5 people be seated on a sofa if there are only 3 seats available 1.65. In how many ways can 7 books be arranged on a shelf if (a) any arrangement is possible, (b) 3 particular books must always stand together, (c) two particular books must occupy the ends
CHAPTER 1 Basic Probability
1.66. How many numbers consisting of five different digits each can be made from the digits 1, 2, 3, . . . , 9 if (a) the numbers must be odd, (b) the first two digits of each number are even 1.67. Solve Problem 1.66 if repetitions of the digits are allowed. 1.68. How many different three-digit numbers can be made with 3 fours, 4 twos, and 2 threes 1.69. In how many ways can 3 men and 3 women be seated at a round table if (a) no restriction is imposed, (b) 2 particular women must not sit together, (c) each woman is to be between 2 men
Combinations
1.70. Evaluate (a) 5C3, (b) 8C4, (c) 10C8. 1.71. For what value of n is 3
n 1C3
7 nC2
1.72. In how many ways can 6 questions be selected out of 10 1.73. How many different committees of 3 men and 4 women can be formed from 8 men and 6 women 1.74. In how many ways can 2 men, 4 women, 3 boys, and 3 girls be selected from 6 men, 8 women, 4 boys and 5 girls if (a) no restrictions are imposed, (b) a particular man and woman must be selected 1.75. In how many ways can a group of 10 people be divided into (a) two groups consisting of 7 and 3 people, (b) three groups consisting of 5, 3, and 2 people 1.76. From 5 statisticians and 6 economists, a committee consisting of 3 statisticians and 2 economists is to be formed. How many different committees can be formed if (a) no restrictions are imposed, (b) 2 particular statisticians must be on the committee, (c) 1 particular economist cannot be on the committee 1.77. Find the number of (a) combinations and (b) permutations of 4 letters each that can be made from the letters of the word Tennessee.