Overview A Monte Carlo experiment is a statistical analysis that uses repetitive random sampling to solve a problem. These experiments are often used in scenarios where physically performing the experiment is not realistic. For example, simulating one million spins on a fair casino Roulette wheel- exactly what you will be doing for this assignment. Roulette is a casino game where a ball is dropped on a spinning wheel. Depending on where the ball randomly lands, the player may win or (more likely) lose money. The term \"Monte Carlo method/experiment\" gets its name from the Monte Carlo Casino in Monaco. Details A European roulette wheel, illustrated above, has slots for the numbers 0through 36 (American roulette wheels have an additional slot: 00). Each number is colored either green, red, or black: 0 D is Green. EUROPEAN 0 1-10 (even numbers), 11-18 (odd numbers), 19-28 (even ROULETTE numbers), and 29-36 (odd numbers) are Black. 0 1-10 (odd numbers), 11-18 (even numbers), 19-28 (odd numbers), and 29-36 (even numbers) are Red. Probabilities Where n represents the number of simulations/spins: The probability of a red number being selected in a single spin is 5 (as shown in the lecture slides). The expected number of red numbers being selected in 2 spins would be 5 + g or; x 2. The expected number of red numbers being selected in 5 spins would be E + E + E + E + E or E x 5. Thus, the 37 37 37 37 37 37 expected number of red numbers appearing in n spins is: 13x 37 n The probability of a black number being selected in a single spin is also g. Thus, the expected number of black numbers appearing in n spins is: Ex 37 " The probability of a green number being selected in a single spin is 5 since there is only one green number (zero). Thus, the expected number of green numbers appearing in n spins is: 1X 37" For ten spins (n = 10): This tells us that we should expect around 5 red and 5 black numbers (maybe 1 green number) in ten spins of the wheel. Now let us consider streaks of red or black numbers and determine what is the longest streak of red and black numbers to expect in n spins. For example, if we spin the wheel 10 times, could we expect the longest streak observed to be red-red-red, red-red-red-red-red, or red-red-red-red-red-red-red-red? Since there an equal number of red and black numbers, only one formula (given below) is needed for determining this: For 10 spins (n = 10): This tells us that we should expect the longest streak of red or black to be around 3 (red-red-red and black-black-black). Simulations You will conduct three separate experiments as part ofthis assignment: - Simulating 100 games . Simulating 1000 games . Simulating 1000000 games Before you begin, calculate the expected number of times red, black, and green numbers should appear in each experiment (like how we figured it out for 10 spins in the previous section of these instructions). Using R, you will oonduct these three experiments and calculate the observed number of times red, black, and green numbers appeared. The roulette_tools.lt file in the Module download contains three functions for getting a vector of spins (the numbers), for getting a vector of the colors of each number in the series, and for calculating the streaks. The roulette_demo.lt program demonstrates how to use the roulette_tools functions for your experiments. You will need to write your own code/function in R to count the total number of "red\Assignment Questions 1. Describe what your two plots show for the IUD-spin simulation. Are they what you expected based on the calculated/expected outcome? 2. Describe what your two plots show for the IDOD-spin simulation. Are they what you expected based on the calculated/expected outcome? 3. Describe what your two plots show for the IDDDDDO-spin simulation. Are they what you expected based on the calculated/expected outcome? 4. Does the "Law of Large Numbers\" appear to hold true in your simulations? Why or why not