HW1_Group2

組員：唐思琪、林書霆、柯炯名、簡佑臻、王婷、胡祐銓

Q3

set.seed(9487)
simulation_runs = 10000
gamble=function(){
  fair_dice = sample(1:6,2,replace = TRUE)
  unfair_dice = sample(1:6,size = 1,replace = TRUE, prob = c(0.15,0.25,0.15,0.15,0.15,0.15))
  result = min(fair_dice,unfair_dice)
}

print(paste0('The expected value of the smallest number of rolling the three dies is: ', mean(replicate(simulation_runs,gamble()))))

[1] "The expected value of the smallest number of rolling the three dies is: 2.0195"

Q4

(a)

set.seed(1287)
t = 1000                     # repeat times
make_sales = function(n=20){ # 20 visits
  results = rep(NA, n)       # initialize table
  # sample the skill of one salesperson
  skill = sample(c('high','low'), 1, prob=c(0.5, 0.5), replace=T)
  if(skill=='high'){
    # given high skill, sample succeed or fail
    results= sample(c('high_succeed','high_fail'), n, prob=c(2/3, 1/3), replace=T)
  }
  else{
    # given low skill, sample succeed or fail
    results= sample(c('low_succeed','low_fail'), n, prob=c(1/3, 2/3), replace=T)
  }
  results
}

salesSim = replicate(t, make_sales())
prob_promoted = sum(colSums(salesSim=='high_succeed'|salesSim=='low_succeed')>=9) / t
print(paste0("The probability of being promoted: ", prob_promoted))

[1] "The probability of being promoted: 0.587"

(b)

• P(promoted | high) = P(promoted & high) / P(high)

prob_high = sum(colSums(salesSim=='high_succeed'|salesSim=='high_fail')!=0) / t
prob_prom_high = sum(colSums(salesSim=='high_succeed')>=9) / t
prom_given_high = prob_prom_high / prob_high
print(paste0("The conditional probability that this person will be promoted if there is high skill: ", round(prom_given_high, 4)))

[1] "The conditional probability that this person will be promoted if there is high skill: 0.9796"

(c)

• P(high | promoted) = P(promoted & high) / P(promoted)

prom_given_high = prob_prom_high / prob_promoted
print(paste0("The conditional probability that this person will be promoted if there is high skill: ", round(prom_given_high, 4)))

[1] "The conditional probability that this person will be promoted if there is high skill: 0.816"

(d)

df = data.frame(matrix(ncol=3, nrow=20))                                # initialize dataframe
rownames(df) = paste0("n=", 1:20)                                       # change row names
colnames(df) = c("P(promoted)", "P(promoted|high)", "P(high|promoted)") # change column names

for (i in 1:nrow(df)) {
  # calculate number of conditions
  no_prob_promoted = sum(colSums(salesSim=='high_succeed'|salesSim=='low_succeed')>=i) 
  no_prob_high = sum(colSums(salesSim=='high_succeed'|salesSim=='high_fail'))
  no_prob_prom_high = sum(colSums(salesSim=='high_succeed')>=i)
  # calculate probilities & append data to df
  df[i,1] = no_prob_promoted / t
  df[i,2] = round(no_prob_prom_high / no_prob_high, 4)
  df[i,3] = round(no_prob_prom_high / no_prob_promoted, 4)
}
df

Q5

set.seed(6781)
simulation_runs = 10000
weds_return = function(runs){
  # simulate the library where the borrower returned the book
  return_library = sample(c(0,1),1,prob = c(0.5,0.5),replace=T) # 1: commerce library, 0: foreign library
  if(any(return_library==1)){ # delay days if return books to a foreign library
    delay = 0 # delay days if return books to a commerce library
  }
  if(any(return_library==0)){ # delay days if return books to a foreign library
    delay = 2
  }
  # simulate the time when the borrower return the book
  return_book = sample(c(4,5,6,7),1,prob = c(0.1,0.2,0.3,0.4),replace=T)
  hello = sum(delay, return_book) # calculate the day that the person will get the book
  hello = hello %% 7
}
result = replicate(simulation_runs,weds_return()) # return the book in how many days 
df = data.frame('days'=c('Thu', 'Fri', 'Sat', 'Mon', 'Tue', 'Wed'), table(result)/simulation_runs )
df[,c(1,3)] # show the return probabilities in each day

print(paste0('The probability that the book is returned on Wednesday is: ', df[6,3])) # answer

[1] "The probability that the book is returned on Wednesday is: 0.1938"