HW6
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.
rm <- 54
wm <- 9
bm <- 75
am <- sum(rm,wm , bm)
rbm <- round((rm/am + bm/am),4) The probability that marbles will be red or blue is 0.9348
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.
gb <- 19
rb <- 20
bb <- 24
yb <- 17
ab <- sum(gb, rb, bb, yb)
prb <- round((rb/ab),4)The probability of getting a red golf ball is 0.25
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. Determine if the following events are independent. Going to the gym. Losing weight.
f <-c(228,79,252,97,72)
m <-c(81,116,215,130,129)
lwp <-c(215,252)
sf <-sum(f)
sm <-sum(m)
stotal <-sum(sf,sm)
slwp <- sum(lwp)
nm <- round((1 -sm/stotal),4)
snlwp <- round((1 - slwp/stotal),4)
nmsnlwp<- nm+snlwp - round((lwp[1]/slwp),4)The probability customer is not male or does not live with parents is 0.7262
Determine if the following events are independent. Going to the gym. Losing weight. Answer: A) Dependent B) Independent
Going to the gym and losing weight are independent. Not all people go to gym to lose weight. Some go to the gym to gain muscle mass and/or just to stay healthy. Also, some people who go to the gym to lose weight don’t always succeed.
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?
v <- 8
c <- 7
t <- 3
vw <- (factorial(v)/(factorial(3) * factorial(v-3))) * (factorial(c)/(factorial(3) * factorial(c-3))) * t
choose(8,3)## [1] 565880 different veggie wraps can be made
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
These events are independent. Running out of gas has no dependency with Liz watching the eveninng news.
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?
s <- 8
c <- 14
a<- factorial(s) / factorial(c-s)There are 56 different ways to appoint members to the cabinet.
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.
r <- 9
o <- 4
g <- 9
a<- sum(r,o,g)
pr <- round(r/a,4)
po <- round(o/a,4)
pg <- round(g/a,4)
prob<-round(((1-pr)*po*(3*pg)),4)The probability of withrawing 4 jellybeans where 0 are red, 1 is orange and 3 are green is 0.1318
Evaluate the following expression. 11! 7!
f<- factorial(11) / factorial(7)The expression gives the number of different permutations if 7 items were select from 11 where order does not matter. In this case the number of permuations are 7920
Describe the complement of the given event. 67% of subscribers to a fitness magazine are over the age of 34.
The complement of 67% of subscribers of a fitness magazine over age 34 would be 100-67% or 33% of subscribers that are under the age of 34
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.)
#Step 1
h <- dbinom(3,4,.5)
t <- dbinom(1,4,.5)
w <- 97
l <- -30
wev <- h*w
lev <- t*l
exp1 <- round((wev + lev),2)
#Step 2
h <- mean(rbinom(559,4,.5)==3)
t <- mean(rbinom(559,4,.5)==1)
wev <- h*w
lev <- t*l
exp2 <- round((wev + lev),2)STEP1 The expected value of 4 coins tosses is 16.75 STEP2 The expected value of 559 coin tosses is 12.91
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.)
#Step 1
h <- dbinom(4,9,.5)
t <- dbinom(5,9,.5)
w <- 23
l <- -26
wev <- h*w
lev <- t*l
exp1 <- round((wev + lev),2)
#Step 2
h <- mean(rbinom(994,9,.5)<=4)
t <- mean(rbinom(994,9,.5)>4)
wev <- h*w
lev <- t*l
exp2 <- round((wev + lev),2)STEP1 The expected value of 9 coins tosses is -0.74 STEP2 The expected value of 994 coin tosses is -1.47
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.
The below matrix provides the probability matrix for answers a-c
l <- .59
t <- .9
ll <- .2
tt <- .8
matrix(c(ll,l, ll*l, ll, 1-l, ll*(1-l), tt, t, t*tt, tt, 1-t, tt*(1-t)), 4,3,byrow =TRUE)## [,1] [,2] [,3]
## [1,] 0.2 0.59 0.118
## [2,] 0.2 0.41 0.082
## [3,] 0.8 0.90 0.720
## [4,] 0.8 0.10 0.080What 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.)
lie<- l*llThe probabiliy that the individual is a liar given they are detected as such is 0.118
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.)
truth<- tt*tThe probabiliy that the individual is a true-teller given they are detected as such is 0.72
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.
lil<- lie + (tt*(1-t))The probabiliy that the individual is either a liar or they are detected as such is 0.198