Governor's School Final 2003 Part 3
Part 3A
1. What are the possible distances (note that the question asks about distances and not displacements; a distance can't be negative) corresponding to a 5-step random walk?
How many possible outcomes are there on 5 flips of a coin? List each possible distance and the corresponding number of ways this distance can occur on a 5-flip random walk.
If the distances corresponding to every possible outcome are averaged, what is the result? Note that the number of distances added must be equal to the total number of possible outcomes.
If you find the square root of the average of the squared distances what do you get?
What is the probability of each distance if you do this random walk with a perfect randomizer?
What do you get if you multiply each possible distance by its probability and how is this result related to one of your previous results?
2. What are the possible distances (note that the question asks about distances and not displacements; a distance can't be negative) corresponding to a 5-step random walk?
How many possible outcomes are there on 6 flips of a coin? List each possible distance and the corresponding number of ways this distance can occur on a 5-flip random walk.
If the distances corresponding to every possible outcome are averaged, what is the result? Note that the number of distances added must be equal to the total number of possible outcomes.
If you find the square root of the average of the squared distances what do you get?
What is the probability of each distance if you do this random walk with a perfect randomizer?
What do you get if you multiply each possible distance by its probability and how is this result related to one of your previous results?
3. What are the possible distances (note that the question asks about distances and not displacements; a distance can't be negative) corresponding to a 5-step random walk?
How many possible outcomes are there on 7 flips of a coin? List each possible distance and the corresponding number of ways this distance can occur on a 5-flip random walk.
If the distances corresponding to every possible outcome are averaged, what is the result? Note that the number of distances added must be equal to the total number of possible outcomes.
If you find the square root of the average of the squared distances what do you get?
What is the probability of each distance if you do this random walk with a perfect randomizer?
What do you get if you multiply each possible distance by its probability and how is this result related to one of your previous results?
4. What are the possible distances (note that the question asks about distances and not displacements; a distance can't be negative) corresponding to a 5-step random walk?
How many possible outcomes are there on 8 flips of a coin? List each possible distance and the corresponding number of ways this distance can occur on a 5-flip random walk.
If the distances corresponding to every possible outcome are averaged, what is the result? Note that the number of distances added must be equal to the total number of possible outcomes.
If you find the square root of the average of the squared distances what do you get?
What is the probability of each distance if you do this random walk with a perfect randomizer?
What do you get if you multiply each possible distance by its probability and how is this result related to one of your previous results?
Part 3B
1.
If the configurations 010, 011, 100 and 111 all result in the middle cell being in the '1' state during the next transition while all other configurations results in the '0' state, then if in a long row of cells only the middle cell is initially in the 1 state, what will be the next 10 configurations of the system?
How well do random do you think the 10 states of the middle cell are? How random do you think the states will be if this rule is continued for another million or so transitions?
What is the binary numbering of this rule?
2. If the configurations 010, 011, 101 and 111 all result in the middle cell being in the '1' state during the next transition while all other configurations results in the '0' state, then if in a long row of cells only the middle cell is initially in the 1 state, what will be the next 10 configurations of the system?
How well do random do you think the 10 states of the middle cell are? How random do you think the states will be if this rule is continued for another million or so transitions?
What is the binary numbering of this rule?
3. If the configurations 001, 011, 100 and 111 all result in the middle cell being in the '1' state during the next transition while all other configurations results in the '0' state, then if in a long row of cells only the middle cell is initially in the 1 state, what will be the next 10 configurations of the system?
How well do random do you think the 10 states of the middle cell are? How random do you think the states will be if this rule is continued for another million or so transitions?
What is the binary numbering of this rule?
4. If the configurations 010, 011, 100 and 101 all result in the middle cell being in the '1' state during the next transition while all other configurations results in the '0' state, then if in a long row of cells only the middle cell is initially in the 1 state, what will be the next 10 configurations of the system?
How well do random do you think the 10 states of the middle cell are? How random do you think the states will be if this rule is continued for another million or so transitions?
What is the binary numbering of this rule?
Part 3C
1. The area under a graph of the normal distribution curve N(z) vs. z, between z = z1 and z = z2, gives the probability that the random variable z will lie between z1 and z2. Sketch a trapezoidal approximation graph of the Normal Distribution function N(z) = 1 / sqrt(2 pi) * e^-(z^2/2) for z = -3, -2, -1, 0, 1, 2 and 3. Label only the altitudes.
Now sketch a 2-interval trapezoidal approximation graph to answer the following question:
If in a certain very difficult class a grade of B is given to anyone whose z score lies between .5 and 1.5, what is the probability that a randomly selected student will get a B? How many of a class of 80 students would we expect to get B's?
2. The area under a graph of the normal distribution curve N(z) vs. z, between z = z1 and z = z2, gives the probability that the random variable z will lie between z1 and z2. Sketch a trapezoidal approximation graph of the Normal Distribution function N(z) = 1 / sqrt(2 pi) * e^-(z^2/2) for z = -3, -2, -1, 0, 1, 2 and 3. Label only the altitudes.
Now sketch a 2-interval trapezoidal approximation graph to answer the following question:
If among 200 cyclers in the Tour de France those whose z scores lie between 1 and 1.8 are expected to finish between 30 and 60 minutes behind the winner, then what proportion will be expected to finish in this range? How many cyclers will this be?
3. The area under a graph of the normal distribution curve N(z) vs. z, between z = z1 and z = z2, gives the probability that the random variable z will lie between z1 and z2. Sketch a trapezoidal approximation graph of the Normal Distribution function N(z) = 1 / sqrt(2 pi) * e^-(z^2/2) for z = -3, -2, -1, 0, 1, 2 and 3. Label only the altitudes.
Now sketch a 2-interval trapezoidal approximation graph to answer the following question:
If newborn female baby whales with z values for fitness which lie between z = .4 and z = 2.2 produce an average of 2 baby whales during their lifetime, then what is the probability that a given baby whale will be within this range? How many baby whales would we expect during the lifetime of 100 newborn female whales?
4. The area under a graph of the normal distribution curve N(z) vs. z, between z = z1 and z = z2, gives the probability that the random variable z will lie between z1 and z2. Sketch a trapezoidal approximation graph of the Normal Distribution function N(z) = 1 / sqrt(2 pi) * e^-(z^2/2) for z = -3, -2, -1, 0, 1, 2 and 3. Label only the altitudes.
Now sketch a 2-interval trapezoidal approximation graph to answer the following question:
If basketball players whose vertical-leap z scores lie between .6 and 1.4 have vertical leaps between 31 inches and 35 inches, then what proportion lie in this range? How many of a random sample of 200 players would be expected to have a vertical leap in this range?
Part 3D
1. If you haven't already done the problem from Part 1 D go back and do it now. If you have completed it then you can skip this problem.
2. Same as problem 1 above.
3. Same as problem 1 above.
4. Same as problem 1 above.
Part 3E
1. If you haven't already done the problem from Part 2 F go back and do it now. If you have completed it then you can skip this problem.
2. Same as problem 1 above.
3. Same as problem 1 above.
4. Same as problem 1 above.
Part 3F
1. If you haven't already done the problem from Part 2 E go back and do it now. If you have completed it then you can skip this problem.
2. Same as problem 1 above.
3. Same as problem 1 above.
4. Same as problem 1 above.