1. A box contains 54 red marbles, 9 white marbles, and 75 blue marbles. If a marble is randomly selected from the box, what is the probability that it is red or blue? Express your answer as a fraction or a decimal number rounded to four decimal places.
marbles_red <- 54
marbles_white <- 9
marbles_blue <- 75

prob_red_or_blue <- (marbles_red + marbles_blue)/(marbles_red + marbles_blue + marbles_white)

round(prob_red_or_blue,4)
## [1] 0.9348
  1. You are going to play mini golf. A ball machine that contains 19 green golf balls, 20 red golf balls, 24 blue golf balls, and 17 yellow golf balls, randomly gives you your ball. What is the probability that you end up with a red golf ball? Express your answer as a simplified fraction or a decimal rounded to four decimal places.
ball_green <- 19
ball_red <- 20
ball_blue <- 24
ball_yellow <- 17

prob_red <- ball_red/(ball_green + ball_red + ball_blue + ball_yellow)

round(prob_red,4)
## [1] 0.25

3 .A pizza delivery company classifies its customers by gender and location of residence. The research department has gathered data from a random sample of 1399 customers. The data is summarized in the table below.

What is the probability that a customer is NOT male or does NOT live with parents? Write your answer as a fraction or a decimal number rounded to four decimal places.

males <- 81 + 116 + 215 + 130 + 129
females <- 228 + 79 + 252 + 97 + 72

males_females <- males + females
prob_not_males <- females/males_females
with_parents_females <- 252
prob_not_males_or_not_with_parents <- (females - with_parents_females)/(males_females)

round(prob_not_males_or_not_with_parents,4)
## [1] 0.3402
  1. Determine if the following events are independent. Going to the gym. Losing weight.

Answer: A) Dependent B) Independent

Going to gym increases likelihood of loosing weight. The likelihood of loosing some weight increases when a person going to gym. The events are dependant.

  1. A veggie wrap at City Subs is composed of 3 different vegetables and 3 different condiments wrapped up in a tortilla. If there are 8 vegetables, 7 condiments, and 3 types of tortilla available, how many different veggie wraps can be made?
choose(8, 3) * choose(7, 3) * choose(3, 1)
## [1] 5880
  1. Determine if the following events are independent. Jeff runs out of gas on the way to work. Liz watches the evening news.

Two events above don’t affect the probability of each other in general. So, they are independent.

Answer: A) Dependent B) Independent

  1. The newly elected president needs to decide the remaining 8 spots available in the cabinet he/she is appointing. If thereare 14 eligible candidates for these positions (where rank matters), how many different ways can the members of the cabinet be appointed?
#it's the case of Permutations without Repetition

#the number of things to choose from
n <- 14

#we choose r of them
r <- 8

factorial(n)/factorial(n-r)
## [1] 121080960
  1. A bag contains 9 red, 4 orange, and 9 green jellybeans. What is the probability of reaching into the bag and randomly withdrawing 4 jellybeans such that the number of red ones is 0, the number of orange ones is 1, and the number of green ones is 3? Write your answer as a fraction or a decimal number rounded to four decimal places.
prob <-choose(9, 0) * choose(4, 1) * choose(9, 3)/choose(22, 4)
round(prob,4)
## [1] 0.0459
  1. Evaluate the following expression. \[11!/7!\]

\[11!/(11-4)!\]

#it's the case of Permutations without Repetition

#the number of things to choose from
n <- 11

#we choose r of them
r <- 4

factorial(n)/factorial(n-r)
## [1] 7920
  1. Describe the complement of the given event. 67% of subscribers to a fitness magazine are over the age of 34.

33% of subscribers to a fitness magazine are the age of 34 and under.

  1. If you throw exactly three heads in four tosses of a coin you win $97. If not, you pay me $30.

Step 1. Find the expected value of the proposition. Round your answer to two decimal places.

#cases with four heads : HHHH, THHH, HTHH, HHTH, HHHT
win_cases <- 5

#all possible cases
#the case of permutations with Repetition
all_cases <- 2^4 #we have 2 possible outcomes(H,T) and we choose 4 of them
all_cases
## [1] 16
prob_win <- win_cases/all_cases
prob_win
## [1] 0.3125
prob_loose <- 1 - prob_win
prob_loose
## [1] 0.6875
#we choose r of them
exp_value <- 97*prob_win - 30*prob_loose
exp_value
## [1] 9.6875

Step 2. If you played this game 559 times how much would you expect to win or lose? (Losses must be entered as negative.)

exp_value *559
## [1] 5415.312
  1. Flip a coin 9 times. If you get 4 tails or less, I will pay you $23. Otherwise you pay me $26.

Step 1. Find the expected value of the proposition. Round your answer to two decimal places.

#all possible cases
all_cases <- 2^9 #we have 2 possible outcomes(H,T) and we choose 9 of them

#cases with one,two,three and four tails cases
#it's the case of combinations without repetition
cases_one_tail <- factorial(9)/(factorial(1)*(factorial(9-1))) 
cases_two_tail <- factorial(9)/(factorial(2)*(factorial(9-2))) 
cases_three_tail <- factorial(9)/(factorial(3)*(factorial(9-3)))
cases_four_tail <- factorial(9)/(factorial(4)*(factorial(9-4))) #9 is the number of things to choose from, and we choose 4 of them,no repetition, order doesn't matter.

cases_no_more_four_tails <- cases_one_tail + cases_two_tail + cases_three_tail + cases_four_tail

prob_win <- cases_no_more_four_tails/all_cases
prob_loose <- 1 - prob_win

#we choose r of them
exp_value <- round(23*prob_win - 26*prob_loose,2)
exp_value
## [1] -1.6

Step 2. If you played this game 994 times how much would you expect to win or lose? (Losses must be entered as negative.)

exp_value*994
## [1] -1590.4
  1. The sensitivity and specificity of the polygraph has been a subject of study and debate for years. A 2001 study of the use of polygraph for screening purposes suggested that the probability of detecting a liar was .59 (sensitivity) and that the probability of detecting a “truth teller” was .90 (specificity). We estimate that about 20% of individuals selected for the screening polygraph will lie.
  1. What is the probability that an individual is actually a liar given that the polygraph detected him/her as such? (Show me the table or the formulaic solution or both.)
prob_liar <- 0.2
sensitivity<- 0.59
specificity <- 0.9
prob_truth_teller <- 1 - prob_liar

false_positve <- 1 - sensitivity
false_negative <- 1 - specificity

prob_liar_detected_liar <- prob_liar * sensitivity
prob_liar_detected_truth_teller <- prob_liar * false_positve
prob_truth_teller_detected_truth_teller <- prob_truth_teller * specificity
prob_truth_teller_detected_liar <- prob_truth_teller * false_negative

prob_detected_liar <- prob_liar_detected_liar + prob_truth_teller_detected_liar
conditional_prob <- prob_liar_detected_liar/prob_detected_liar

round(conditional_prob,4)
## [1] 0.596
  1. What is the probability that an individual is actually a truth-teller given that the polygraph detected him/her as such? (Show me the table or the formulaic solution or both.)
prob_detected_truth_teller <- prob_liar_detected_truth_teller + prob_truth_teller_detected_truth_teller
conditional_prob <- prob_truth_teller_detected_truth_teller/prob_detected_truth_teller

round(conditional_prob,4)
## [1] 0.8978
  1. What is the probability that a randomly selected individual is either a liar or was identified as a liar by the polygraph? Be sure to write the probability statement.
prob_liar + prob_truth_teller_detected_liar
## [1] 0.28