2. If 8 rooks (castles) are randomly placed on a chessboard, compute the proba 3. If...
2. If
8
rooks (castles) are randomly placed on a chessboard, compute the
proba
3. If
20
people, including Alice and Bob, are arranged in a
circle, what is the probability that
Alice and Bob are NOT next to each other?
Homework Answers
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2. If 8 rooks (castles) are randomly placed on a chessboard, compute the proba 3. If...
4. If 8 rooks (castles) are randomly placed on a chessboard, compute the probability that none...
4. If 8 rooks (castles) are randomly placed on a chessboard, compute the probability that none of the rooks can capture any of the others. That is, compute the probability that no row or file contains more than one rook.
4. If 8 rooks (castles) are randomly placed on a chessboard, compute the probability that none...
4. If 8 rooks (castles) are randomly placed on a chessboard, compute the probability that none of the rooks can capture any of the others. That is, compute the probability that no row or file contains more than one rook. 5. A pair of dice is rolled until a sum of either 5 or 7 appears. Find the probability that a 5 occurs first. (Hint: Let F denote the event that a 5 occurs on the nth roll and no...
Eight rooks are placed randomly on a chess board. What is the probability that none of the rooks can capture any of the other rooks? Translation for those who are not familiar with chess: pick 8 unit...
Eight rooks are placed randomly on a chess board. What is the probability that none of the rooks can capture any of the other rooks? Translation for those who are not familiar with chess: pick 8 unit squares at random from an 8 × 8 square grid. What is the probability that no two chosen squares share a row or a column? Hint. You can think of placing the rooks both with or without order, both approaches work.
4. Eight rooks are placed randomly on a chess board. What is the probability that none...
4. Eight rooks are placed randomly on a chess board. What is the probability that none of the rooks can capture any of the other rooks? (In non-chess terms: Randomly pick 8 unit squares from an 8 x 8 square grid. What is the probability that no two squares share a row or a column?) Hint: How many choices do you have to place rooks in the first row? After you have made your choice, how many choices do you...
Question 2: Part 1: Show that it is not possible to choose a uniform positive integer...
Question 2: Part 1: Show that it is not possible to choose a uniform positive integer at random. (In other words, we cannot define a probability measure on the positive integers that can be considered uniform). Hint: What would be the probability of choosing a particular number? Part 2: Eight rooks are placed randomly on a chess board. What is the probability that none of the rooks can capture any of the other rooks? Translation for those unfamiliar with chess:...
Suppose you are the Lord Commander of the Night’s Watch. You have 3 identical armies and 3 different castles (Castle Black (1), Eastwatch (2), and the Shadow Tower (3)) to defend. A possible defense a...
Suppose you are the Lord Commander of the Night’s Watch. You have 3 identical armies and 3 different castles (Castle Black (1), Eastwatch (2), and the Shadow Tower (3)) to defend. A possible defense arrangement is a placement of armies at the 3 castles. Castles can be left without an army to defend it. a. Recall that each defense arrangement corresponds to a binary string of length 5 with 2 ones. Consider this alternative random strategy: Construct a length 5...
Suppose you are the Lord Commander of the Night’s Watch. You have 3 identical armies and 3 different castles (Castle Black (1), Eastwatch (2), and the Shadow Tower (3)) to defend. A possible defense a...
Suppose you are the Lord Commander of the Night’s Watch. You have 3 identical armies and 3 different castles (Castle Black (1), Eastwatch (2), and the Shadow Tower (3)) to defend. A possible defense arrangement is a placement of armies at the 3 castles. Castles can be left without an army to defend it. a. Recall that each defense arrangement corresponds to a binary string of length 5 with 2 ones. Consider this alternative random strategy: Construct a length 5...
answer all questions 7. (4 points) What is the probability that an integer from 1 to...
answer all questions
7. (4 points) What is the probability that an integer from 1 to 1000, inclusive, chosen at random, is a multiple of 3 or a multiple of 7? 8. Eight people, including Joe and Ellen, are going to the movies and will sit in a row of 8 chairs. (a) (3 points) In how many ways can the eight people seat themselves? (b) (4 points) In how many ways can they seat themselves if Joe and Ellen...
4. Consider a chessboard, shown below. At starting at the square al (the lower right hand...
4. Consider a chessboard, shown below. At starting at the square al (the lower right hand corner), you can go one step up or one step right at each move. The procedure stops until the point h8 (the upper right corner) is reached. a) How many different paths from al to h8 are possible? (b) How many differnt paths from al to h8 are possible if each path must pass through the square e4? 5. If 10 new teachers are...
4. Suppose you are the Lord Commander of the Night’s Watch. You have 3 identical armies and 3 different castles (Castle Black (1), Eastwatch (2), and the Shadow Tower (3)) to defend. A possible defens...
4. Suppose you are the Lord Commander of the Night’s Watch. You
have 3 identical armies and 3 different castles (Castle Black (1),
Eastwatch (2), and the Shadow Tower (3)) to defend. A possible
defense arrangement is a placement of armies at the 3 castles.
Castles can be left without an army to defend it. Recall that using
the stars and bars counting technique, there are 3+2 2 = 5 2
different defense arrangement
(c) (9 points) Recall that each...
4. If 8 rooks (castles) are randomly placed on a chessboard, compute the probability that none of the rooks can capture any of the others. That is, compute the probability that no row or file contains more than one rook.
4. If 8 rooks (castles) are randomly placed on a chessboard, compute the probability that none of the rooks can capture any of the others. That is, compute the probability that no row or file contains more than one rook. 5. A pair of dice is rolled until a sum of either 5 or 7 appears. Find the probability that a 5 occurs first. (Hint: Let F denote the event that a 5 occurs on the nth roll and no...
4. Eight rooks are placed randomly on a chess board. What is the probability that none of the rooks can capture any of the other rooks? (In non-chess terms: Randomly pick 8 unit squares from an 8 x 8 square grid. What is the probability that no two squares share a row or a column?) Hint: How many choices do you have to place rooks in the first row? After you have made your choice, how many choices do you...
Question 2: Part 1: Show that it is not possible to choose a uniform positive integer at random. (In other words, we cannot define a probability measure on the positive integers that can be considered uniform). Hint: What would be the probability of choosing a particular number? Part 2: Eight rooks are placed randomly on a chess board. What is the probability that none of the rooks can capture any of the other rooks? Translation for those unfamiliar with chess:...
answer all questions
7. (4 points) What is the probability that an integer from 1 to 1000, inclusive, chosen at random, is a multiple of 3 or a multiple of 7? 8. Eight people, including Joe and Ellen, are going to the movies and will sit in a row of 8 chairs. (a) (3 points) In how many ways can the eight people seat themselves? (b) (4 points) In how many ways can they seat themselves if Joe and Ellen...
4. Consider a chessboard, shown below. At starting at the square al (the lower right hand corner), you can go one step up or one step right at each move. The procedure stops until the point h8 (the upper right corner) is reached. a) How many different paths from al to h8 are possible? (b) How many differnt paths from al to h8 are possible if each path must pass through the square e4? 5. If 10 new teachers are...
4. Suppose you are the Lord Commander of the Night’s Watch. You
have 3 identical armies and 3 different castles (Castle Black (1),
Eastwatch (2), and the Shadow Tower (3)) to defend. A possible
defense arrangement is a placement of armies at the 3 castles.
Castles can be left without an army to defend it. Recall that using
the stars and bars counting technique, there are 3+2 2 = 5 2
different defense arrangement
(c) (9 points) Recall that each...
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