Instruction: Write a function in R that performs a Monte Carlo simulation to estimate the value of Pi.
Context: This question assesses the candidate's understanding of statistical simulation techniques and their ability to implement them in R for solving complex problems.
Official answer available
Preview the opening of the answer, then unlock the full walkthrough.
At its core, the Monte Carlo simulation relies on randomness to simulate processes and compute estimates. To estimate Pi, we can simulate the throwing of darts at a square board that has a quarter circle inscribed within it. If we randomly throw darts at this board, the ratio of darts that land inside the quarter circle to those that land inside the square will approximate Pi/4. This is because the area of the quarter circle is Pi*R^2/4 (where R is the radius), and the area of the square is R^2, leading to a ratio of Pi/4 when we divide the area of the quarter circle by the area of the square.
To implement this in R, we'll write a function that performs the simulation:...