Answers in R to frequently discussed statistical questions. Mostly derived from Coursera `An Intuitive Introduction to Probability`

stat_problems.r

#In a room of n people what is the probability of someone having a bday today
#This can be solved by 1 - prob of someone not havng bday today
# prob of someone not havng bday today = 364/365
# for n people this probabilty is same everyone. Hence these probability gets multiplied.
# 1 - above probability would the answer
#For n=253, thes probability reaches 50%
birthday_today = function(n){
  1 - ((364/365)^n)
}

#In a room of n people, prob of two (or more) people having a same bday
#If some one has a bday then other person having a different bday is 364/365
#For the next person it would be 363/365
#For the third person it would be 362/364 and so on..
#Tese probs gets multiplied which gives us the total prob of of n people having different bday
# 1-above  probability would the answer
# For n=23 prob of two people in a room of 23 people having same bday reaches 50%
same_bday = function(n){
  prob = 1 #bday for first person (it can be anyday hence it would be one 365/365)
  for(i in 2:n){
    prob = prob * ((366-i)/365)
  }
  1-prob #contains prob of two people having same bday
}

#Above thing can be simulated randomly
birthday_prob = function(n = 23, times = 1000){
  sim = lapply(1:times, function(i){
    length(which(duplicated(sample(x = 1:365, size = n, replace = TRUE))))
  })
  hist(unlist(sim))
}

#Monty Hall problem
#As Karl says during the lecture, it's an infamous (or famous) one named after geme show host Monty Hall
#Lets simmulate

monty_hall= function(user_switches = TRUE){
  doors = c("D1", "D2", "D3")

  outcome = NULL

  user_door = sample(doors, size = 1) #Contestsent chooses a door
  #message("Contestent choice: ",user_door)
  correct_door = sample(doors, size = 1) #Correct door with a car in it
  #message("Correct door: ",correct_door)

  #Now monty opens a false door (He is not allowed to open the correct one)
  monty_opens = doors[!doors %in% c(user_door, correct_door)]
  if(length(monty_opens) > 1){
    monty_opens = sample(x = monty_opens, size = 1)
  }
  #message("Monty opens: ",monty_opens)
  remaining_doors = doors[!doors %in% monty_opens]
  #message("Remaining doors: ", paste(remaining_doors, collapse = ", "))

  #user_switches = sample(x = c(TRUE, FALSE), size = 1)

  if(user_switches){
    user_door = remaining_doors[!remaining_doors %in% user_door]
    #message("Contestent wishes to swicth. His new door: ", user_door)
  }else{
    #message("Contestent does not wishes to swicth. He sticks to ", user_door)
  }

  if(user_door == correct_door){
    outcome = "WON"
  }else{
    outcome = "LOST"
  }

  #message("Outcome: ", outcome)
  outcome
}

#Lets simulate above game for 1000 times
game_outcomes_when_door_swicthed = c()
for(i in 1:1000){
  game_outcomes_when_door_swicthed = c(game_outcomes, monty_hall(user_switches = TRUE))
}
print(prop.table(table(game_outcomes_when_door_swicthed)))

game_outcomes_when_door_not_swicthed = c()
for(i in 1:1000){
  game_outcomes_when_door_not_swicthed = c(game_outcomes_when_door_not_swicthed, monty_hall(user_switches = FALSE))
}
print(prop.table(table(game_outcomes_when_door_not_swicthed)))

#It seems ~57% of time contenstent WON the game when he chose to switch the door, whereas he WON only ~32% of the time when did not swicth