605 HW10

Smith is in jail and has 1 dollar; he can get out on bail if he has 8 dollars. A guard agrees to make a series of bets with him. If Smith bets A dollars, he wins A dollars with probability .4 and loses A dollars with probability .6. Find the probability that he wins 8 dollars before losing all of his money if:

I will run 500 simulations for determining the probability.

(a) he bets 1 dollar each time (timid strategy)

p = vector("numeric", 500)
bets = vector("numeric", 500)

for (i in 1:500) { 
  
  #start = 1, win 8
  d = 1
  num_of_bet = 0
  while (!(d == 0 || d >= 8)) {
    
    num_of_bet = num_of_bet + 1

    winLose = runif(1, min=0, max=1) 
  
    if(winLose <= 0.4) {
      d = d + 1 #win
    }
    else {
      d = d - 1 #lose
    }
    
  }
  
  #this simulation result:
  if (d == 8) { #if win, set to 1
    p[i] = 1 
  }
  else { #else, set to 0
    p[i] = 0
  }
  
  bets[i] = num_of_bet
  
}

#store the probablity
timid_strategy_p = sum(p)/500

paste("With with timid strategy, the probability that smith win 8 dollors: ", timid_strategy_p)

## [1] "With with timid strategy, the probability that smith win 8 dollors:  0.026"

(b) he bets, each time, as much as possible but not more than necessary to bring his fortune up to 8 dollars (bold strategy)

p = vector("numeric", 500)
bets = vector("numeric", 500)

for (i in 1:500) { 
  
  #start = 1, win 8
  d = 1
  num_of_bet = 0
  while (!(d == 0 || d >= 8)) {
    
    num_of_bet = num_of_bet + 1

    winLose = runif(1, min=0, max=1) 
  
    if(winLose <= 0.4) {
      d = d * 2 #win
    }
    else {
      d = 0 #lose
    }
    
  }
  
  #this simulation result:
  if (d == 8) { #if win, set to 1
    p[i] = 1 
  }
  else { #else, set to 0
    p[i] = 0
  }
  
  bets[i] = num_of_bet
  
}

bold_strategy_p = sum(p)/500

paste("With bold strategy, the probability that smith win 8 dollors: ",bold_strategy_p )

## [1] "With bold strategy, the probability that smith win 8 dollors:  0.06"

(c) Which strategy gives Smith the better chance of getting out of jail?

if (bold_strategy_p > timid_strategy_p){
    paste("The bold strategy gives Smith the better chance of getting out of jail by probability: ", (bold_strategy_p - timid_strategy_p) )

}

## [1] "The bold strategy gives Smith the better chance of getting out of jail by probability:  0.034"

if(bold_strategy_p <= timid_strategy_p) {
    paste("The timid strategy gives Smith the better chance of getting out of jail by probability: ", abs(bold_strategy_p - timid_strategy_p) )
}