Problem $1$

Consider a server farm composed of $20$ identical servers that can handle requests from various clients. A

request from a client is assigned to one of the servers completely at random $($with uniform probability$).$

Consider $10$ requests submitted to this farm. Answer the following questions $[2$ pts each$]$:

$1.$ What is the probability that all $10$ requests go the same machine?

$2.$ What is the probability that each request goes to a different machine?

Problem $2$

Consider a web$-$server that receives web requests at an average rate $\backslash$lambda of $2$ requests per second that follows

a Poisson distribution. Answer the following questions: $[2$ pts each$]$

$($a$)$ What$$s the probability that it would receive $4$ requests in a second?

$($b$)$ What$$s the probability that no requests will arrive for $3$ seconds?

$($c$)$ Consider one second and assume that $2$ requests already arrived in $0\mathrm{.}5$ second, what is the probability

that $2$ requests will arrive in the next $0\mathrm{.}5$ seconds $?$

Problem $3$

Using a pseudo random number generation function $($e$.$g$.,$ rand$()$ in C or other equivalent functions in other

languages$)$ that generates uniformly distributed random numbers, write three functions that generate the

following: $[3$ pts each$]$

$($a$)100$ uniformly distributed integers between $0$ and $99.$

$($b$)100$ uniformly distributed floating numbers between $0\mathrm{.}25$ and $0\mathrm{.}5.$

$($c$)100$ numbers in which the number $1$ is produced with probability $0\mathrm{.}5,$ the number $2$ with probability $0\mathrm{.}2,$

otherwise a floating number between $3$ and $4.$

Please submit your code and a sample output. You can use any language as long as it compiles and runs on