$($based on $6\mathrm{.}12$ from UsingR$)$ We can use probability to estimate a value

of $.$ Pick a point at random from a square of side length $2$ centered at the

origin. Inside the square draw a circle of radius $1.$ The probability that

the point is inside the circle is equal to the area of the circle divided by the

area of the square: $(\backslash$pi $12)/(22)=\backslash$pi $/4.$ We can simulate the probability

and then multiply by $4$ to estimate $\backslash$pi $.$ The simulation could be done with

following command:

n $=1000$; x $=$ runif$($n$,1,1)$

y $=$runif$($n$,1,1)$;sum$($x$2+$ y$2<1)/$n

Do a simulation to estimate $\backslash$pi $.$ What do you get? Use the binomial model

and the known value of $\backslash$pi to find the standard deviation of the random

variable you estimated.