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3) Consider two balanced dices with each of the six faces marked 1 to 6. In...
1) Prove that if loga x = y, then lo8b x = (loga x)(logb a). Plot...
1) Prove that if loga x = y, then lo8b x = (loga x)(logb a). Plot loga n for 1 S ns 100 when a 2, e and 10.(10-points) 2) Prove that (b) Ση-112-n(n+ 1 )(2n+ 1 )/6 k+1) (1-x) if X Hint: Induction - Discrete Math (15-points) 3) Consider two balanced dices with each of the six faces marked 1 to 6. In one single throw of these two dice, list all the possible outcomes. Compute the average value...
Part I - Throwing a Dice with Six Faces (32 points) Consider a dice with six...
Part I - Throwing a Dice with Six Faces (32 points) Consider a dice with six faces, i.e. a standard dice. The possible outcomes after throwing the dice once are 1, 2,3, 4,5,6 (a) Assume that the dice is thrown once. Let w represents the outcome. Explain why P(w-i) for i = i..6, when the dice is fair (b) The fair dice is thrown twice. Find the probability of occurance of at least one 6 The fair dice is thrown...
1. Consider a six-faced die whose faces have respective colours red, red, red, blue, blue, green....
1. Consider a six-faced die whose faces have respective colours red, red, red, blue, blue, green. (a) Describe the random variable (i.e. possible outcomes and probability distribution) of a fair die throw (i.e. one where each face ends up on top with equal likelihood). (b) What is the entropy of the random variable of part (a)? (c) Suppose two identical such dice are thrown simultaneously. What is the probability that i. both dice come up red? ii. the dice come...
1. Two dices are thrown (a) List the elements of the sample space. (b) List the...
1. Two dices are thrown (a) List the elements of the sample space. (b) List the outcomes that define the following events. E: At least one of the dice rolls on 6. he same number . F: Both dice roll on t . G: The sum of the dice is odd H: The number on the first die is larger than the number on the second die. (c) Explain whether the following events are mutually exclusive or not (i) E...
A six-sided dice is loaded in a way that each even face is twice as likely...
A six-sided dice is loaded in a way that each even face is twice as likely as each odd face. All even faces are equally likely, as are all odd faces. a. Construct a probabilistic model for a single roll of this dice. i.e. construct a table (or a rule) that lists all the possible outcomes and the probability associated with each. b. Find the probability that, on a single roll, the outcome is at least 3.
5.A six-sided dice is loaded in a way that each even face is twice as likely...
5.A six-sided dice is loaded in a way that each even face is twice as likely as each odd face. All even faces are equally likely, as are all odd faces. a. Construct a probabilistic model for a single roll of this dice. i.e. construct a table (or a rule) that lists all the possible outcomes and the probability associated with each. b. Find the probability that, on a single roll, the outcome is at least 3
Show up your work 5. A six-sided dice is loaded in a way that each even...
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5. A six-sided dice is loaded in a way that each even face is twice as likely as cach odd face. All even faces are equally likely, as are all odd faces. Construct a probabilistic model for a single roll of this dice. i.c. construct a table (or a rule) that lists all the possible outcomes and the probability associated with each. Find the probability that, on a single roll, the outcome is at least 3....
3-(35 points) Two standard 6-sided dice with 6 faces having values (1,2,3,4,5,6) are thrown and the...
3-(35 points) Two standard 6-sided dice with 6 faces having values (1,2,3,4,5,6) are thrown and the face values on the top face of cach die are observed. Let E be the event that the maximum of the dice face values is odd Let F be the event that none of the dice lands on the value 1. Let G be the event that the product is less or equal to 6. a) Determine the probabilities of all events: E, F,...
1. (25 points 6+6+6+7) For a fair throw of six 6-sided dice (as in the game...
1. (25 points 6+6+6+7) For a fair throw of six 6-sided dice (as in the game Farkle): (a) Find the probability of 6 of a kind ANSWER (b) Find the probability of two different triplets. ANSWER (c) Find the probability of a four-of-a-kind and a different pair. ANSWER: (d) Now suppose the six dice are each of a different color and all placed an urn. After having chosen 4 with replacement, find the probability that two or more colors are...
Two six-sided dice will be rolled once and the numbers (number of dots) on each dice...
Two six-sided dice will be rolled once and the numbers (number of dots) on each dice is to be recorded. Define events Еґ the sum of the two dice is 1,1-2,3, , 12. a. List all the outcomes in the Sample Space. Calculate the probability of each event, El, E2, , E12. c. Let A be the event "the sum of the two dice is greater than 6" Calculate P(E10|A) and P(A|E10)
1) Prove that if loga x = y, then lo8b x = (loga x)(logb a). Plot loga n for 1 S ns 100 when a 2, e and 10.(10-points) 2) Prove that (b) Ση-112-n(n+ 1 )(2n+ 1 )/6 k+1) (1-x) if X Hint: Induction - Discrete Math (15-points) 3) Consider two balanced dices with each of the six faces marked 1 to 6. In one single throw of these two dice, list all the possible outcomes. Compute the average value...
Part I - Throwing a Dice with Six Faces (32 points) Consider a dice with six faces, i.e. a standard dice. The possible outcomes after throwing the dice once are 1, 2,3, 4,5,6 (a) Assume that the dice is thrown once. Let w represents the outcome. Explain why P(w-i) for i = i..6, when the dice is fair (b) The fair dice is thrown twice. Find the probability of occurance of at least one 6 The fair dice is thrown...
1. Two dices are thrown (a) List the elements of the sample space. (b) List the outcomes that define the following events. E: At least one of the dice rolls on 6. he same number . F: Both dice roll on t . G: The sum of the dice is odd H: The number on the first die is larger than the number on the second die. (c) Explain whether the following events are mutually exclusive or not (i) E...
A six-sided dice is loaded in a way that each even face is twice as likely as each odd face. All even faces are equally likely, as are all odd faces. a. Construct a probabilistic model for a single roll of this dice. i.e. construct a table (or a rule) that lists all the possible outcomes and the probability associated with each. b. Find the probability that, on a single roll, the outcome is at least 3.
5.A six-sided dice is loaded in a way that each even face is twice as likely as each odd face. All even faces are equally likely, as are all odd faces. a. Construct a probabilistic model for a single roll of this dice. i.e. construct a table (or a rule) that lists all the possible outcomes and the probability associated with each. b. Find the probability that, on a single roll, the outcome is at least 3
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5. A six-sided dice is loaded in a way that each even face is twice as likely as cach odd face. All even faces are equally likely, as are all odd faces. Construct a probabilistic model for a single roll of this dice. i.c. construct a table (or a rule) that lists all the possible outcomes and the probability associated with each. Find the probability that, on a single roll, the outcome is at least 3....
3-(35 points) Two standard 6-sided dice with 6 faces having values (1,2,3,4,5,6) are thrown and the face values on the top face of cach die are observed. Let E be the event that the maximum of the dice face values is odd Let F be the event that none of the dice lands on the value 1. Let G be the event that the product is less or equal to 6. a) Determine the probabilities of all events: E, F,...
1. (25 points 6+6+6+7) For a fair throw of six 6-sided dice (as in the game Farkle): (a) Find the probability of 6 of a kind ANSWER (b) Find the probability of two different triplets. ANSWER (c) Find the probability of a four-of-a-kind and a different pair. ANSWER: (d) Now suppose the six dice are each of a different color and all placed an urn. After having chosen 4 with replacement, find the probability that two or more colors are...
Two six-sided dice will be rolled once and the numbers (number of dots) on each dice is to be recorded. Define events Еґ the sum of the two dice is 1,1-2,3, , 12. a. List all the outcomes in the Sample Space. Calculate the probability of each event, El, E2, , E12. c. Let A be the event "the sum of the two dice is greater than 6" Calculate P(E10|A) and P(A|E10)
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