$1\mathrm{.}4$ Problems Part $4$

$1.$ Buffon's needle problem is another way to estimate the value of $\backslash$pi

with random numbers. The goal in this Monte Carlo estimate of $\backslash$pi

is to create a ratio that is close to $3\mathrm{.}1415926...$ similar to the example with darts points lying inside$/$outside a unit circle inside a unit square.

Buffon's needle for parallel

lines

In this Monte Carlo estimation, you only need to know two values:

the distance from line $0,$ x$=[0,1]$

the orientation of the needle, $\backslash$theta $=[0,2\backslash$pi $]$

The y$-$location does not affect the outcome of crosses line $0$ or not crossing line $0.$

a$.$ Generate $100$ random x and theta values remember $\backslash$theta $=[0,2\backslash$pi $]$

b$.$ Calculate the x locations of the $100$ needle ends e$.$g$.$ xend$=$x$\backslash$pm cos$\backslash$theta

$\_$since length is unit $1.$

c$.$ Use np$.$logical$\_$and to find the number of needles that have minimum xend min$<0$

and maximum xend max$>0$

$.$ The ratio xend min$<0$ and xend max$>0$number of needles$=2\backslash$pi

for large values of number of needles

$.$

$2.$ Build a random walk data set with steps between dx$=$dy$=1/2$ to $1/2$ m

$.$ If $100$ particles take $10$ steps, calculate the number of particles that move further than $0\mathrm{.}5$ m$.$

Bonus: Can you do the work without any for$-$loops? Change the size of dx and dy to account for multiple particles.