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.

#54 red marbles, 9 white marbles, and 75 blue marbles
MAR = 54 + 9 + 75
#Total Marbles
MAR

## [1] 138

#Probability its red
(54/MAR)

## [1] 0.3913043

#Probability blue
(75/MAR)

## [1] 0.5434783

#Probability its red or blue
round( ((54/MAR) + (75/MAR)),4)

## [1] 0.9348

2. 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.

#Total balls :
BAL = 19+20+24+17
BAL

## [1] 80

#Red balls = 20
RBAL = 20 
#Probability of Red ball: 
RBAL/BAL

## [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.

#Total number of males and females

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

## [1] 1399

#Per compliment rule The Saample that satisfies either "not male" or "not living with parents" are males living with their parents That is 215

P<-round(((SAM - 215)/SAM),4)
P

## [1] 0.8463

4. Determine if the following events are independent. Going to the gym. Losing weight. Answer: A) Dependent B) Independent

Independent. From our experience they are dependent events but without further/additional mathematical data we cannot deduce they are Dependent. The loss of event could be dependent on other factors like health issues or weather.

5. 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?

# Ways you can choose 3 vegetables from 8 vegetable options:- choose(8,3)
# Ways you can choose 3 condiments from 7 condiment options:- choose(7,3)
# Ways you can choose a tortilla from 3 tortilla options:- choose(3,1)
# Total possible combinations of choosing veggie wraps

choose(8,3) * choose(7,3) * choose(3,1)

## [1] 5880

6. Determine if the following events are independent. Jeff runs out of gas on the way to work. Liz watches the evening news. Answer: A) Dependent B) Independent

Independent. The events are in no way related.

7. 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?

\[P(n, r) = \frac{n!}{(n - r)!}\]

factorial (14) / factorial (14-8)

## [1] 121080960

8. 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 greenones is 3? Write your answer as a fraction or a decimal number rounded to four decimal places.

Totbeans <- 9 + 4 + 9

4 is drawn out of Totbeans is \[C^{Totbeans}_{4}\] or can be represented as choose(Totbeans,4)

round(((choose(9,3) * choose(4,1)) / choose(Totbeans,4)),4)

## [1] 0.0459

9. Evaluate the following expression. 11! / 7!

factorial(11)/factorial(7)

## [1] 7920

10. 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 34 or younger.

11. 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.

win_prob = pbinom(3, size=4, prob=0.5) - pbinom(2, size=4, prob=0.5)
loss_prob = 1-win_prob
#Expected value
EV = round(((97 * win_prob) - (30*loss_prob)),2)
EV

## [1] 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.)

If you played the game 559 time you will make

paste("$",(559 * EV))

## [1] "$ 978.25"

12. 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.

win_prob = pbinom(4, size=9, prob=0.5)
loss_prob = 1-win_prob
#Expected value
EV = round(((23 * win_prob) - (26*loss_prob)),2)
EV

## [1] -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.)

If you played the game 994 time you will make

paste("$",994 * EV)

## [1] "$ -1491"

13. 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.

a. 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.)

# Probability that someone selected for polygraph is lying
prob_liar <- 0.2
# Probability that the sensitivity test will detect a liar
prob_det <- 0.59
# Probability that someone who's lying will be detected as a liar
prob_liar * prob_det

## [1] 0.118

b. 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.)

# Probability that someone selected for polygraph is being truthful
prob_tt <- 0.8
# Probability that the specificity test will detect truth-telling
prob_det <- 0.9
# Probability that someone who's lying will be detected as a liar
prob_tt * prob_det

## [1] 0.72

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.

# Probability that someone selected for polygraph is lying
prob_liar <- 0.2
# Probability that the sensitivity test will detect a liar
prob_det <- 0.59
# Chance that someone who's lying will be detected as a liar
prob_liar * prob_det

## [1] 0.118

# Probability that a randomly selected individual is either a liar or was identified as a liar by the polygraph 
# P(liar) + P(detected as liar) - P(A liar and is detected as a liar)
prob <- prob_liar + prob_det - (prob_liar * prob_det)
prob

## [1] 0.672

data-605-week6-Assignment

Ashish Kumar

03/04/2020

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.

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.

4. Determine if the following events are independent. Going to the gym. Losing weight. Answer: A) Dependent B) Independent

5. 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?

6. Determine if the following events are independent. Jeff runs out of gas on the way to work. Liz watches the evening news. Answer: A) Dependent B) Independent

7. 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?

9. Evaluate the following expression. 11! / 7!

10. Describe the complement of the given event. 67% of subscribers to a fitness magazine are over the age of 34.

11. 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.

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

12. 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.

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

a. 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.)

b. 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.)

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.