Question

Estimating π. Using random numbers can accomplish many tasks. For example, it is possible to estimate π using Monte Carlo methods. To get an estimate, you place random points in the square [0, 1] × [0, 1] and estimate π/4 as the ratio of the points that landed inside the unit circle to the total number of points (because the ratio of the area of the circle to the area of the square is π/4). Here is what a few test runs and the corresponding estimates for π might look like with the number of points equal to 100, 1000, and 10,000 respectively?

Answer 1

Answer:

Estimations of π/4:

100 points: 0.75

1,000 points: 0.768

10,000 points: 0.7819

Step-by-step explanation:

To get an estimate of π/4 you can place random points in the square [0, 1] × [0, 1] and estimate it as the ratio of the points that landed inside the unit circle to the total number of points (because the ratio of the area of the circle to the area of the square is π/4).

We do it for 100 xy points and we get:

Point inside the circle area = 75

Estimation of π/4 = 0.75

$\pi/4\approx=\dfrac{\text{points inside circle area}}{\text{total points}}=\dfrac{75}{100}=0.75$

We do it for 1,000 xy points and we get:

Point inside the circle area = 768

Estimation of π/4 = 0.768

$\pi/4\approx=\dfrac{\text{points inside circle area}}{\text{total points}}=\dfrac{768}{1000}=0.768$

If we do it fo 10,000 xy points, we get

Point inside the circle area = 768

Estimation of π/4 = 0.768

$\pi/4\approx=\dfrac{\text{points inside circle area}}{\text{total points}}=\dfrac{7819}{10000}=0.7819$

The value of π/4 (4 decimals) is 0.7854.

The simulation gets more precise with the increase in the number of points.

The spreadsheet and the graphs are attached.