Analysis of switching strategy for Monty Hall problem

Overview and summary

The Monty Hall problem is a brain teaser, in the form of a probability puzzle (Gruber, Krauss and others), loosely based on the American television game show Let’s Make a Deal and named after its original host, Monty Hall. The problem was originally posed in a letter by Steve Selvin to the American Statistician in 1975 (Selvin 1975a), (Selvin 1975b).

Our objective in this assignment is to write an R-script to simulate the conditional probabilities and variances of the distributions for winning a car vs. a goat for the Monty Hall Problem. Include the cases of switching or not switching. Compare these cases to determine the best strategy.

Note: Following packages are required to run the below report. plyr ggplot2 *scales

rm(list=ls())
require (plyr)

## Loading required package: plyr

require(ggplot2)

## Loading required package: ggplot2

require(scales)

## Loading required package: scales

Data Preparation:

MontyHall.Problem.Simulation <- function (noofgates, r, strategy)
{
  dfmontymatrix <- data.frame(matrix(0 ,ncol = 6, nrow = r))
  cnames <- c("Gatechosen", "PrizeGate","MontySelection","Strategy", "Result", "Noofsimulations")
  colnames(dfmontymatrix) <- cnames
  #noofgates <- 1:3

  dfmontymatrix[, "Noofsimulations"] <- r  # No of simulations.
  for (ri in 1: r)
  {
    indexprize <- sample(noofgates, 1, replace = TRUE) 
    Gatechosen <- sample(noofgates, 1, replace = TRUE)
    
    if(indexprize != Gatechosen)
      {
      MontySelection <- noofgates[-c(indexprize, Gatechosen)]
    } else 
      { 
          MontySelection <- sample(noofgates[-c(indexprize, Gatechosen)], 1)
      } 
    
    dfmontymatrix[ri, "Gatechosen"] <- Gatechosen
    dfmontymatrix[ri, "PrizeGate"] <- indexprize
    dfmontymatrix[ri, "MontySelection"] <- MontySelection
    dfmontymatrix[ri, "Strategy"] <- sample(strategy, 1, replace = TRUE) 
   
    if (dfmontymatrix[ri,"Strategy"] == "stay")
    {
      if (dfmontymatrix[ri,"Gatechosen"] == dfmontymatrix[ri, "PrizeGate"])
      {
        dfmontymatrix[ri, "Result"] = "win"
      } 
      else 
      {
        dfmontymatrix[ri, "Result"] = "loss"
      } 
    } else
    {
      
      if (dfmontymatrix[ri,"Gatechosen"] != dfmontymatrix[ri, "PrizeGate"])
      {
        dfmontymatrix[ri, "Result"] = "win"
      } 
      else 
      {
        dfmontymatrix[ri, "Result"] = "loss"
      }   
    }
  }
  
  return (with (dfmontymatrix, table(Noofsimulations, Strategy, Result)))
  
}

Data simulation:

Lets simulate the data with datapoints (1000, 10000, 20000)

r <- c( 1000, 10000, 20000)
noofgates <-1:3
strategy <- c("stay", "switch") 
sampledata <- lapply(r, function(r) MontyHall.Problem.Simulation(noofgates, r, strategy))
final.output <- ldply (sampledata, data.frame)
final.output$Noofsimulations <- as.integer(as.character(final.output$Noofsimulations))
final.output <- mutate(final.output, 
                       Percentage = percent(Freq/ Noofsimulations))

print (final.output)

##    Noofsimulations Strategy Result Freq Percentage
## 1             1000     stay   loss  318      31.8%
## 2             1000   switch   loss  159      15.9%
## 3             1000     stay    win  178      17.8%
## 4             1000   switch    win  345      34.5%
## 5            10000     stay   loss 3347      33.5%
## 6            10000   switch   loss 1641      16.4%
## 7            10000     stay    win 1625      16.2%
## 8            10000   switch    win 3387      33.9%
## 9            20000     stay   loss 6706      33.5%
## 10           20000   switch   loss 3306      16.5%
## 11           20000     stay    win 3324      16.6%
## 12           20000   switch    win 6664      33.3%

Visualization:

ggplot(final.output, aes(Result,Freq, fill = Strategy ))+
  geom_bar( stat = "identity")+
  geom_text(aes(label = Percentage))+
  facet_wrap(~Noofsimulations, scales = "free" )+
  labs(title = "Analysis of strategies involved for Monty Hall problem ", y = "No of Simulations", x = "Strategy Result")

Lets do only for strategy = “switch”

strategy <-  "switch" 

sampledata <- lapply(r, function(r) MontyHall.Problem.Simulation(noofgates, r, strategy))

final.output <- ldply (sampledata, data.frame)
final.output$Noofsimulations <- as.integer(as.character(final.output$Noofsimulations))
final.output <- mutate(final.output, 
                       Percentage = percent(Freq/ Noofsimulations))

ggplot(final.output, aes(Result,Freq, fill = Strategy ))+
  geom_bar( stat = "identity")+
  geom_text(aes(label = Percentage))+
  facet_wrap(~Noofsimulations, scales = "free" )+
  labs(title = "Analysis of switching strategy for Monty Hall problem ", y = "No of Simulations", x = "Strategy Result")

Lets do only for strategy = “stay”

strategy <-  "stay" 

sampledata <- lapply(r, function(r) MontyHall.Problem.Simulation(noofgates, r, strategy))

final.output <- ldply (sampledata, data.frame)
final.output$Noofsimulations <- as.integer(as.character(final.output$Noofsimulations))
final.output <- mutate(final.output, 
                       Percentage = percent(Freq/ Noofsimulations))

ggplot(final.output, aes(Result,Freq, fill = Strategy ))+
  geom_bar( stat = "identity")+
  geom_text(aes(label = Percentage))+
  facet_wrap(~Noofsimulations, scales = "free" )+
  labs(title = "Analysis of not switching strategy for Monty Hall problem ", y = "No of Simulations", x = "Strategy Result")

Conclusion:

Based on the above visualizations, Switching strategy has more chances to win than staying.