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.

Answer: Probability that the marble selected is a red or blue marble = (number of red or blue marbles) / total number of marbles

library(dplyr)
## 
## Attaching package: 'dplyr'
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
((54 + 75) / (54 + 75 + 9)) %>% round(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.

Answer: Probability of getting a red ball = (number of red balls available) / (total number of balls)

(20 / (19 + 20 + 24 + 17)) %>% round(4)
## [1] 0.25
  1. 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.

Gender and Residence of Customers Males Females Apartment 81 228 Dorm 116 79 With Parent(s) 215 252 Sorority/Fraternity House 130 97 Other 129 72

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.

Answer: P(customer is not a male or does not live with parents) - As this is an or, that means this event is same as : either the customer is a female OR the customer does not live with parents, which is equivalent to: all customers from the table MINUS males living with parents = 1399 - 215

((1399 - 215) / 1399) %>% round(4)
## [1] 0.8463

If the question was for: probability that a customer is neither male nor does the customer live with parents, then this event would be equivalent to: all female customers MINUS females living with parents

((228 + 79 + 97 + 72) / 1399) %>% round(4)
## [1] 0.3402

  1. Determine if the following events are independent. Going to the gym. Losing weight Answer: These are dependent events.

  2. 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?

Answer:

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.

Answer: Tese are independent events.

  1. The newly elected president needs to decide the remaining 8 spots available in the cabinet he/she is appointing. If there are 14 eligible candidates for these positions (where rank matters), how many different ways can the members of the cabinet be appointed?

Answer: As the rank matters here, this will be the Permutation instead of the Combination.

factorial(14) / factorial(14 - 8)
## [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.

Answer: Total number of ways of withdrawing 4 jellybeans:

(9 + 4 + 9) %>% choose(4)
## [1] 7315

Number of ways to withdraw : 1 orange and 3 green ones:

choose(4, 1) * choose(9, 3)
## [1] 336

Probability of withdrawing 1 orange and 3 green ones = (Number of ways to withdraw : 1 orange and 3 green ones) / (Total number of ways of withdrawing 4 jellybeans)

((choose(4, 1) * choose(9, 3)) / ((9 + 4 + 9) %>% choose(4))) %>% round(4)
## [1] 0.0459
  1. Evaluate the following expression.

11! / 7!

Answer: 11! / 7! = (11.10.9.8.7!) / 7! = 11.10.9.8 = 7920

Alternatively using R:

factorial(11) / factorial(7)
## [1] 7920
  1. Describe the complement of the given event. 67% of subscribers to a fitness magazine are over the age of 34.

Answer: The remaining 33% subscribers of the magazine are 34 or below. As both add up to give 100% or probability of 1, the above is the complement of the even given in the question.

  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.

Answer: Let us first find the total number of possible outcomes when 4 coins are tossed. This will give the total number.

total_number_of_ways_11 <- 2^4
total_number_of_ways_11
## [1] 16

Now, the number of ways exactly 3 heads can come:

number_of_ways_3_heads_11 <- choose(4, 3)
number_of_ways_3_heads_11
## [1] 4

Now, probablity of getting exactly 3 heads:

probability_3_heads_11 <- number_of_ways_3_heads_11 / total_number_of_ways_11
probability_3_heads_11
## [1] 0.25

Value of the proposition:

value_of_proposition_11 <- probability_3_heads_11 * 97 + (1 - probability_3_heads_11) * (-30)
value_of_proposition_11
## [1] 1.75

Hence, the value of this proposition is +1.75

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

Answer: Expected win or loss = value of each possible game (value proposition) X number of games played

expected_win_or_loss_11 <- value_of_proposition_11 * 559
expected_win_or_loss_11
## [1] 978.25

Hence I am expected to win $978.25 if I play 559 games

  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.

Answer: Similar to exercise 11, we will first calculate the total number of possible outcomes:

total_number_of_ways_12 <- 2^9
total_number_of_ways_12
## [1] 512

Now we will find the possible number of ways to have 4 tails or less, which will be = Number of ways to have 0 Tails + 1 Tails + 2 Tails + 3 Tails + 4 Tails

number_of_ways_4orless_tails <- 1 + choose(9, 1) + choose(9, 2) + choose(9, 3) + choose(9, 4)
number_of_ways_4orless_tails
## [1] 256
probability_4orless_tails <- number_of_ways_4orless_tails / total_number_of_ways_12
probability_4orless_tails
## [1] 0.5
value_of_proposition_12 <- probability_4orless_tails * 23 + (1 - probability_4orless_tails) * (-26)
value_of_proposition_12
## [1] -1.5

Value of proposition = -1.5

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

Answer:

expected_win_or_loss_12 <- value_of_proposition_12 * 994
expected_win_or_loss_12
## [1] -1491

Value of expected_win_or_loss_12 is -1491, which is a negative value. So, I am expected to lose 1491, if I play this game 994 times.

  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.)

Answer: Assigning the given values to variables:

p_liar = 0.2
p_detect_liar_a_liar = 0.59
p_detect_truthteller_a_truthteller = 0.9
p_truthteller = 1 - p_liar

Calculating the absolute values now:

p_liar_detected_liar = p_liar * p_detect_liar_a_liar
p_liar_detected_truthteller = p_liar * (1 - p_detect_liar_a_liar)
p_truthteller_detected_liar = p_truthteller * (1 - p_detect_truthteller_a_truthteller)
p_truthteller_detected_truthteller = p_truthteller * p_detect_truthteller_a_truthteller

Now calculating this answer:

p_liar_given_detected_liar = (p_liar_detected_liar / (p_liar_detected_liar + p_truthteller_detected_liar)) %>% round(4)

p_liar_given_detected_liar
## [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.) Answer:
p_truthteller_given_detected_truthteller = (p_truthteller_detected_truthteller / (p_truthteller_detected_truthteller + p_liar_detected_truthteller)) %>% round(4)

p_truthteller_given_detected_truthteller
## [1] 0.8978

C. 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. Answer:

p_either_liar_or_detected_liar = (p_liar + p_truthteller_detected_liar) %>% round(4)

p_either_liar_or_detected_liar
## [1] 0.28