Let's use a simulation to investigate how often we would win the game if we decided to switch doors half the time $($i$.$e$.,$ with probability $0\mathrm{.}50)$ when playing many rounds of the game. The simulation shown in the template is used to simulate $1,000$ rounds of the game in which you initially select Door #$1.$ The vector switch records whether you decide to switch, while the vector wins records whether you win the car. The value $1$ in switch represents deciding to switch and the value $1$ in wins represents a win. The sample$()$ function is used so that switch has value $1$ half the time.

b$)$ The code stops at specifying the outcomes for when the car is behind Door #$1.$ Complete the code by specifying how outcomes should be assigned if the car is behind Door #$2$ or Door #$3.$

c$)$ Run the simulation and view the results.

i$.$ Based on the simulation results, what is the approximate overall probability of winning with this strategy of switching doors $50\%$ of the time?

ii$.$ Based on the simulation results, is it a better strategy to always switch doors or always

not switch? Explain your answer, referencing relevant numerical results.

d$)$ Using an algebraic approach, calculate the probability of winning by always switching doors.

Let Ws represent the event of winning by always switching doors. It may be helpful to define the events of the car being behind Door #$1,$ Door #$2,$ or Door #$3$ as D$,,$ D$2,$ and D$3,$ respectively, and to suppose that you initially select Door #$1.$

replicates

num.doors

#create empty vectors to store results

switch rep$($NA$,$ replicates$)$

wins rep$($NA$,$ replicates$)$

#set seed for a pseudo$-$random sample

set.seed$(2020)$

#simulate rounds of the game

for$($k in $1$:replicates$)\{$switch$[$k sample$($c$(0,1),$ size $=1)$ #choose whether to switch or notif$($car$,$ door $==1)\{$ if$($switch$[$k$]==0)\{$wins$[$k#outcomes if car is behind door $2$#outcomes if car is behind door $3\}$

#view the resultse$)$ Consider a modification of the game in which you are asked to select from one of four doors, where one door leads to a car and the others lead to goats $($all other aspects of the game remain the same; Monty will only open one door and that door must lead to a goat$).$ Calculate the probability of winning by always switching doors and compare this to the probability of winning by never switching doors.