Chapter 9.7, Problem 64E

a.

To determine

Find the probability that the coin covers a vertex of one of the squares on the grid.

Answer to Problem 64E

  $\frac{\pi }{4}$

Explanation of Solution

Calculation:

There is a grid of squares on which coin is dropped. The coin is of diameter $d$ and square is of $d$ units on a side

Let us try to find the probability that the coin covers a vertex of one of the squares on the grid

Let $C$ be the point on the paper where the centre of the coin lands.

The coin is going to cover a vertex if $C$ is less than $d$ units away from a vertex

In each square, the points that are less than $d$ units away from a vertex from four quarter-circles of radius $\frac{d}{2}$

Adding to an Area of $=\pi \times {\left(\frac{d}{2}\right)}^2$

  $=\frac{\pi \times d^2}{4}$

Total area of each square is ${\left(d\right)}^2$

We see that $\frac{\pi }{4}$ of each square, hence $\frac{\pi }{4}$ of the entire paper surface, consists of points that are less than $d$ units away from a vertex

It follows that $C$ has probability $\frac{\pi }{4}$ is being less than $d$ units away from a vertex

Hence, that is the coin has probability $\frac{\pi }{4}$ of covering a vertex

b.

To determine

Perform the experiment $100$ times and use the results to approximate $\pi$ .

Answer to Problem 64E

  $\pi \approx \frac{n}{25}$

Explanation of Solution

Calculation:

Let us perform the experiment many times

We know that probability that the coin covers a vertex of one of the squares is $\frac{\pi }{4}$

If we drop the coin on the paper $100$ times, the coin should end up covering a vertex about

  $\begin{array}{l}=100\times \frac{\pi }{4}\\ =25\times \pi \end{array}$

Hence, we observe that the coin covers vertex $n$ times out of $100$ ,

  $\pi \approx \frac{n}{25}$