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1.78. Calculate (a) 6 C3, (b) a 11 b, (c) (8C2)(4C3) > 12C5. 4 y)4, (c) (x 2 xb .
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1.79. Expand (a) (x
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1.80. Find the coefficient of x in ax
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1.81. Find the probability of scoring a total of 7 points (a) once, (b) at least once, (c) twice, in 2 tosses of a pair of fair dice.
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CHAPTER 1 Basic Probability
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1.82. Two cards are drawn successively from an ordinary deck of 52 well-shuffled cards. Find the probability that (a) the first card is not a ten of clubs or an ace; (b) the first card is an ace but the second is not; (c) at least one card is a diamond; (d) the cards are not of the same suit; (e) not more than 1 card is a picture card ( jack, queen, king); (f) the second card is not a picture card; (g) the second card is not a picture card given that the first was a picture card; (h) the cards are picture cards or spades or both. 1.83. A box contains 9 tickets numbered from 1 to 9, inclusive. If 3 tickets are drawn from the box 1 at a time, find the probability that they are alternately either odd, even, odd or even, odd, even. 1.84. The odds in favor of A winning a game of chess against B are 3:2. If 3 games are to be played, what are the odds (a) in favor of A winning at least 2 games out of the 3, (b) against A losing the first 2 games to B 1.85. In the game of bridge, each of 4 players is dealt 13 cards from an ordinary well-shuffled deck of 52 cards. Find the probability that one of the players (say, the eldest) gets (a) 7 diamonds, 2 clubs, 3 hearts, and 1 spade; (b) a complete suit. 1.86. An urn contains 6 red and 8 blue marbles. Five marbles are drawn at random from it without replacement. Find the probability that 3 are red and 2 are blue. 1.87. (a) Find the probability of getting the sum 7 on at least 1 of 3 tosses of a pair of fair dice, (b) How many tosses are needed in order that the probability in (a) be greater than 0.95 1.88. Three cards are drawn from an ordinary deck of 52 cards. Find the probability that (a) all cards are of one suit, (b) at least 2 aces are drawn. 1.89. Find the probability that a bridge player is given 13 cards of which 9 cards are of one suit.
Miscellaneous problems
1.90. A sample space consists of 3 sample points with associated probabilities given by 2p, p2, and 4p value of p. 1. Find the
1.91. How many words can be made from 5 letters if (a) all letters are different, (b) 2 letters are identical, (c) all letters are different but 2 particular letters cannot be adjacent 1.92. Four integers are chosen at random between 0 and 9, inclusive. Find the probability that (a) they are all different, (b) not more than 2 are the same. 1.93. A pair of dice is tossed repeatedly. Find the probability that an 11 occurs for the first time on the 6th toss. 1.94. What is the least number of tosses needed in Problem 1.93 so that the probability of getting an 11 will be greater than (a) 0.5, (b) 0.95 1.95. In a game of poker find the probability of getting (a) a royal flush, which consists of the ten, jack, queen, king, and ace of a single suit; (b) a full house, which consists of 3 cards of one face value and 2 of another (such as 3 tens and 2 jacks); (c) all different cards; (d) 4 aces.
CHAPTER 1 Basic Probability
1.96. The probability that a man will hit a target is 2. If he shoots at the target until he hits it for the first time, find 3 the probability that it will take him 5 shots to hit the target. 1.97. (a) A shelf contains 6 separate compartments. In how many ways can 4 indistinguishable marbles be placed in the compartments (b) Work the problem if there are n compartments and r marbles. This type of problem arises in physics in connection with Bose-Einstein statistics. 1.98. (a) A shelf contains 6 separate compartments. In how many ways can 12 indistinguishable marbles be placed in the compartments so that no compartment is empty (b) Work the problem if there are n compartments and r marbles where r n. This type of problem arises in physics in connection with Fermi-Dirac statistics. 1.99. A poker player has cards 2, 3, 4, 6, 8. He wishes to discard the 8 and replace it by another card which he hopes will be a 5 (in which case he gets an inside straight ). What is the probability that he will succeed assuming that the other three players together have (a) one 5, (b) two 5s, (c) three 5s, (d) no 5 Can the problem be worked if the number of 5s in the other players hands is unknown Explain. 1.100. Work Problem 1.40 if the game is limited to 3 tosses. 1.101. Find the probability that in a game of bridge (a) 2, (b) 3, (c) all 4 players have a complete suit.
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