Assignment 5 covers Probability and Simulation.

This assignment provides a more challenging dice game called "Can't Stop". Note this assignmenthas no assumptions(the dice are fair, and the rules are clear).Hence, the report should focus on how you model or simulate the game and the results that you obtain from that model.

Problem description

"Can't Stop" is a strategy and luck game. The game rules are provided here.https://www.ultraboardgames.com/cant-stop/game-rules.php --- this is the summary from the website : ## Can't Stop Game Rules

### Overview In *Can't Stop*, players aim to win three of the eleven number columns on the game board. This is achieved by rolling four dice and forming two pairs, which determine the columns where players can place white cones (runners). Players can continue rolling as long as they can place or move at least one runner. If they can't, they lose everything gained during that turn. The game's addictive nature is evident in its nameplayers often can't stop rolling, hoping for just one more move.

### Components - **1 Game Board:** Featuring 11 numbered columns (2 to 12). - **4 Red Dice** - **3 White Traffic Cones (runners):** Used by all players. - **44 Player Traffic Cones:** 11 for each of the 4 players. - **Rulebook**

### Object of the Game The goal is to advance your cones to the top squares of the columns. The first player to occupy the top squares of three columns wins.

### Setup 1. Place the game board in the center. 2. Each player selects a color and takes 11 cones. 3. The youngest player starts with the 4 dice and 3 white cones.

### Game Play 1. **Starting the Turn:** The youngest player rolls all 4 dice and forms two pairs. 2. **Placing Runners:** - The sums of the pairs determine the columns where runners can be placed. - If two pairs can place runners, the player must place both. - Alternatively, the player can form pairs to place only one runner.

#### Example: - **Roll:** The player rolls: (let's assume 2, 3, 4, 6). - **Options:** - (2+3) and (4+6): Place runners on columns 5 and 10. - (2+4) and (3+6): Place runners on columns 6 and 9. - (2+6) and (3+4): Place runners on columns 8 and 7.

3. **Continuing the Turn:** - The player may keep rolling as long as they can place or move runners. - They can choose to stop voluntarily at any time and place their colored cones where the runners are. - If unable to place or move runners, the turn ends, and they lose any progress made during that turn.

4. **Stopping Voluntarily:** - Place one of your cones on each square reached by a runner. - If a runner lands in a column where the player already has a cone, move the cone up to the new square.

5. **Forced Stop:** - If the player can't place or move a runner, the turn ends, and no progress is saved.

### Winning a Column - A runner reaching the top square of a column secures that column for the player. - The player places their cone on the top square, and all other cones in that column are removed. - No player, including the owner, can place a runner in this column again.

### End of the Game - The game ends immediately when a player secures the top squares of three columns. - This player is declared the winner.

You can find examples and descriptions of the game online.

Start the assignments by becoming familiar with the rules. You may ask for clarification on the discussion board.

At each turn, there are two decisions that need to be made:

• how should you divide up the four dice you've rolled into two pairs?
• And after you've moved the white runners, should you stop? or throw the four dice again?

Once you have placed the 3 white runners on the board, the chances of busting increase. To know what is the risk of keeping playing, we want to calculate the probability of advancing (or busting) given the current 3 locations of the white runners: columns c1, c2 and c3.

Part 1 (70% marks)

Validate the following probabilities of losing the white runners (busting) for the following two situations:

When columns are (3, 8, 11), the chance of advancing is 76% and the chance of busting is 24%. (or 0.24151 to be exact)

When columns are columns are (2, 4, 11) the chance of advancing is 63% and the chance of busting is 37%. (or 0.36574 to be exact)

Part 2 (30% marks)

We want to understand the "ideal" number of throws, by working out the chance of busting after 1,2,3 or 4 throws. Simulate the game from the starting positions a large number of times to provide a reasonable approximation for these values.

submission:

• a pdf file with the format specified underAssignment Instructions. Failure to label a section will result in 0 marks : Assignment 5

Criteria	Ratings	Pts
This criterion is linked to a learning outcomePart1 DescriptionExplanation of part1 approach		50 pts
This criterion is linked to a learning outcomePart1 Resultbust/win % are validated		20 pts
This criterion is linked to a learning outcomePart2 DescriptionApproach to part2		20 pts
This criterion is linked to a learning outcomePart2 Result		10 pts
Total points: 100