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
TM = 54 + 9 + 75
#Total Marbles
TM
## [1] 138
#Probability its red or blue
round( ((54/TM) * (75/TM)),4)
## [1] 0.2127
  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.
#Total number of balls :
TB = 19+20+24+17
TB
## [1] 80
#Number of red balls = 20
RB = 20 
#Probability of getting a red ball: 
RB/TB
## [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.
  m <- c(81, 116, 215, 130, 129)
f <- c(228, 79, 252, 97, 72)
class <- c("Apartment", "Dorm", "With Parent(s)", "Sorority/Fraternity House", "Other")
df <- data.frame(class, m, f)
names(df) <- c("Class", "Males", "Females")
df
##                       Class Males Females
## 1                 Apartment    81     228
## 2                      Dorm   116      79
## 3            With Parent(s)   215     252
## 4 Sorority/Fraternity House   130      97
## 5                     Other   129      72
#Probability that a customer is not Male and does not live with their parents
# Using Compliment rule

#Number of males living with parents =
MP = 215
#Total males and females
TMF = sum(df$Males) + sum(df$Females)

PE= MP/TMF
#The compliment of that event will be  People that are not Male and does not live with their parents.
1-PE
## [1] 0.8463188

  1. Determine if the following events are independent. Going to the gym. Losing weight.

    Answer: A) Dependent B) Independent

    ANS: Yes the the events are independent. example: Suppose we have two person A and B. Person A goes daily to gym whereas Person B goes only 2 days a week. Its independent of each person’s workout time and several other parameters.

  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?

\[ Total different veggie wraps = (Selecting 3 vegetable out of 8)*(selecting 3 condiments out of 7)*(select 1 tortilla out of 3)\]

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: A) Dependent B) Independent

    The events could be Dependent provided there is a connecting relation between two events.

  2. 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?

# 14 eligible candidates
# 8 spots available
# Number of ways members can be appointed is:
choose(14,8) 
## [1] 3003
  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.

ANS: Total = 9 Red + 4 Orange + 9 Green =22

   4 is with drawn out of 22 = 22C4 ways

   Select 4 jelly beans in a way that there are 0 red and 3 green and 1 orange.
(choose(9,3) * choose(4,1)) / choose(22,4)   
## [1] 0.04593301
  1. Evaluate the following expression. 11! / 7!
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.

    34 % of subscribers are below age 34.

  2. 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.

ANS: 3 heads in 4 tosses = \[(1/2)^3 =1/8 \] Probability of win \[ $97 = 1/8 \] Probability of lose \[ $30 =1-1/8 =7/8 \] Expected value of the proposition

( (1/8) * 97) + ( (7/8)*(-30))
## [1] -14.125

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

#Playing this game 559 times

559 * (-14.25)
## [1] -7965.75
  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.

#Expected value. Using binomial

pwin<-pbinom(4,size=9, prob=0.5)

pwin
## [1] 0.5
expval<-round(23*pwin - 26*(1-pwin))

expval
## [1] -2

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

# Playing 994 times
# Lose is:
994 * expval
## [1] -1988
  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.)

L=Liar, D=Detected as Liar

\[ P(L|D) = P(D \cap L)/P(D) = P(D|L)P(L) / ( P(D|L)P(L) + P(D|L`)P(L`) ) \] \[ 0.59*0.2 / ( 0.59*0.2 + 0.1*0.8) \]

0.59*0.2 / ( 0.59*0.2 + 0.1*0.8)
## [1] 0.5959596
  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.)

\[ P(L`|D`) = P(D' \cap L)/P(D') = P(D`|L`)P(L`) / ( P(D|L)P(L) + P(D`|L`)P(L`) ) \] \[ 0.9*0.8 / ( (1 - 0.59)*0.2 + 0.9*0.8) \]

 0.9*0.8 / ( (1 - 0.59)*0.2 + 0.9*0.8)
## [1] 0.8977556
  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.

\[ P(L) + P(L`|D) \]

\[ P(L`|D) = P(D \cap L`) / P( D) \]

\[ P(D|L`) P(X`) / P(D|X)P(X) + P(D|L`) P(X`) \]

 B = 0.1*0.8 / ( (0.59*0.2) +( 0.1*0.8))

 0.2 + B
## [1] 0.6040404