## Suppose You Are Doing A Random Walk See Previous Problem On The Blocks Of A City Street

1. Draw the top levels of a structure chart for a program having the following main function.

printIntro()

length, width = getDimensions() amtNeeded = computeAmount(length,width) printReport(length, width, amtNeeded)

2. Write an expression using either random or randrange to calculate the following.

A random number representing the roll of a six-sided die

• A random number representing the sum resulting from rolling two six-sided dice

3. Revise the racquetball simulation so that it keeps track of results for best of n game matches.

4. Revise the racquetball simulation to take shutouts into account. Your updated version should report for both players the number of wins, percentage of wins, number of shutouts, and percentage of wins that are shutouts.

5. Design and implement a simulation of the game of volleyball. Normal volleyball is played like racquetball, in that a team can only score points when it is serving. Games are played to 15, but must be won by at least two points.

6. College volleyball is now played using rally scoring. In this system, the team that wins a rally is awarded a point, even if they were not the serving team. Games are played to a score of 21. Design and implement a simulation of volleyball using rally scoring.

7. Design and implement a system that compares regular volleyball games to those using rally scoring. Your program should be able to investigate whether rally scoring magnifies, reduces, or has no effect on the relative advantage enjoyed by the better team.

8. Design and implement a simulation of some other racquet sport (e.g. tennis or table tennis).

9. Craps is a dice game played at many casinos. A player rolls a pair of normal six-sided dice. If the initial roll is 2, 3 or 12, the player loses. If the roll is 7 or 11, the player wins. Any other initial roll causes the player to "roll for point." That is, the player keeps rolling the dice until either rolling a 7 or re-rolling the value of the initial roll. If the player re-rolls the initial value before rolling a 7, it's a win. Rolling a 7 first is a loss.

Write a program to simulate multiple games of craps and estimate the probability that the player wins. For example, if the player wins 249 out of 500 games, then the estimated probability of winning is 249 500 0 498

10. Blackjack (twenty-one) is a casino game played with cards. The goal of the game to draw cards that total as close to 21 points as possible without going over. All face cards count as 10 points, aces count as 1 or 11, and all other cards count their numeric value.

The game is played against a dealer. The player tries to get closer to 21 (without going over) than the dealer. If the dealer busts (goes over 21) the player automatically wins (provided the player had not already busted). The dealer must always take cards according to a fixed set of rules. The dealer takes cards until he or she achieves a total of at least 17. If the dealer's hand contains an ace, it will be counted as 11 when that results in a total between 17 and 21 inclusive; otherwise, the ace is counted as 1.

Write a program that simulates multiple games of blackjack and estimates the probability that the dealer will bust.

11. A blackjack dealer always starts with one card showing. It would be useful for a player to know the dealer's bust probability (see previous problem) for each possible starting value. Write a simulation program that runs multiple hands of blackjack for each possible starting value (ace-10) and estimates the probability that the dealer busts for each starting value.

12. Monte Carlo techniques can be used to estimate the value of pi. Suppose you have a round dart board that just fits inside of a square cabinet. If you throw darts randomly, the proportion that hit the dart board vs. those that hit the cabinet (in the corners not covered by the board) will be determined by the relative area of the dart board and the cabinet. If n is the total number of darts randomly thrown (that land within the confines of the cabinet), and h is the number that hit the board, it is easy to show that

Write a program that accepts the "number of darts" as an input and then performs a simulation to estimate p. Hint: you can use 2*random() - 1 to generate the x and y coordinates of a random point inside a 2x2 square centered at (00. The point lies inside the inscribed circle if x2 + y2 < 1.

13. Write a program that performs a simulation to estimate the probability of rolling five-of-a-kind in a single roll of five six-sided dice.

14. A random walk is a particular kind of probablistic simulation that models certain statistical systems such as the Brownian motion of molecules. You can think of a one-dimensional random walk in terms of coin flipping. Suppose you are standing on a very long straight sidewalk that extends both in front of and behind you. You flip a coin. If it comes up heads, you take a step forward; tails means to take a step backward.

Suppose you take a random walk of n steps. On average, how many steps away from the starting point will you end up? Write a program to help you investigate this question.

15. Suppose you are doing a random walk (see previous problem) on the blocks of a city street. At each "step" you choose to walk one block (at random) either forward, backward, left or right. In n steps, how far do you expect to be from your starting point? Write a program to help answer this question.

16. Write a graphical program to trace a random walk (see previous two problems) in two dimensions. In this simulation you should allow the step to be taken in any direction. You can generate a random direction as an angle off of the x axis.

The new x and y positions are then given by these formulas.

The program should take the number of steps as an input. Start your walker at the center of a 100x100 grid and draw a line that traces the walk as it progresses.

17. (Advanced) Here is a puzzle problem that can be solved with either some fancy analytic geometry (calculus) or a (relatively) simple simulation.

Suppose you are located at the exact center of a cube. If you could look all around you in every direction, each wall of the cube would occupy | of your field of vision. Suppose you move toward one of the walls so that you are now half-way between it and the center of the cube. What fraction of your field of vision is now taken up by the closest wall? Hint: use a Monte Carlo simulation that repeatedly "looks" in a random direction and counts how many times it sees the wall.

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