output.txt

> df
   r_correct r_deserved_grade r_guess r_to_pass       r_prob
1          0             0.00      40        23  0.001288255
2          1             0.25      39        22  0.002026315
3          2             0.50      38        21  0.003423564
4          3             0.75      37        20  0.006220180
5          4             1.00      36        19  0.012010891
6          5             1.25      35        18  0.024172832
7          6             1.50      34        17  0.049693691
8          7             1.75      33        16  0.102629931
9          8             2.00      32        15  0.210340863
10         9             2.25      31        14  0.424088520
11        10             2.50      30        13  0.835777412
12        11             2.75      29        12  1.601882250
13        12             3.00      28        11  2.972850586
14        13             3.25      27        10  5.320058407
15        14             3.50      26         9  9.142002221
16        15             3.75      25         8 15.019021384
17        16             4.00      24         7 23.479922196
18        17             4.25      23         6 34.761123278
19        18             4.50      22         5 48.498637220
20        19             4.75      21         4 63.495827014
21        20             5.00      20         3 77.801516427
22        21             5.25      19         2 89.288350784
23        22             5.50      18         1 96.617373932

prob.R

number_of_options_per_question = 4
passing_grade = 5.75 / 10
total_questions = 40 
points_per_question = 1 / total_questions


prob <- function(to_pass, total_questions) {
  total = 0
  for(i in seq(1, to_pass)) {
    total = total + dbinom(i, total_questions, 1/number_of_options_per_question)
  }
  
  return(1-total)
  
}



r_correct <- c()
r_guess <- c()
r_prob <- c()
r_to_pass <- c()
r_deserved_grade <- c()

# 'how many correct' represents how many questions the student actually knows and doesn't have to guess
for(how_many_correct in seq(0,(total_questions*passing_grade)-1)) {
  
  # how many questions does she still need to pass?
  questions_to_pass = passing_grade / points_per_question - how_many_correct
  if(questions_to_pass<=0)
    next
  
  # how many questions can she guess?
  how_many_to_guess = total_questions - how_many_correct
  
  pass_prob <- prob(questions_to_pass, how_many_to_guess)
  
  r_correct <- c(r_correct, how_many_correct)
  r_deserved_grade <- c(r_deserved_grade, how_many_correct/total_questions * 10)
  r_to_pass <- c(r_to_pass, questions_to_pass)
  r_guess <- c(r_guess, how_many_to_guess)
  r_prob <- c(r_prob, pass_prob*100)

}

df<-data.frame(r_correct, r_deserved_grade, r_guess, r_to_pass, r_prob)