#1. There are 540 identical plastic chips numbered 1 through 540 in a box. What is the probability of reaching into the box and randomly drawing the chip numbered 505? Express your answer as a fraction or a decimal number rounded to four decimal places.
round(1/540,4)
## [1] 0.0019
#2. Write out the sample space for the given experiment. Separate your answers using commas.
#When deciding what you want to put into a salad for dinner at a restaurant, you will choose one of the following extra toppings: asparagus, #cheese. Also, you will add one of following meats: eggs, turkey. Lastly, you will decide on one of the following dressings: French, #vinaigrette. (Note: Use the following letters to indicate each choice: A for asparagus, C for cheese, E for eggs, T for turkey, F for #French, and V for vinaigrette.)
tpng <- c(rep("A", 4), rep("C", 4))
meat <- c(rep("E", 2), rep("T", 2), rep("E", 2), rep("T", 2))
dressng <- c("F","V","F","V","F","V","F","V")
df <- data.frame(tpng, meat, dressng)
df
## tpng meat dressng
## 1 A E F
## 2 A E V
## 3 A T F
## 4 A T V
## 5 C E F
## 6 C E V
## 7 C T F
## 8 C T V
#3. A card is drawn from a standard deck of 52 playing cards. What is the probability that the card will be a heart and not a face card? Write your answer as a fraction or a decimal number rounded to four decimal places.
round(10/52,4)
## [1] 0.1923
#4. A standard pair of six-sided dice is rolled. What is the probability of rolling a sum less than 6? Write your answer as a fraction or a decimal number rounded to four decimal places.
round(10/36 ,4)
## [1] 0.2778
#5. A pizza delivery company classifies its customers by gender and location of residence. The research department has gathered data from a random sample of 2001 customers. The data is summarized in the table below.
male <- c(233, 159, 102, 220, 250)
fem <- c(208, 138, 280, 265, 146)
class <- c("Apartment", "Dorm", "With Parent(s)", "Sorority/Fraternity House", "Other")
df <- data.frame(class, male, fem)
names(df) <- c("Class", "Males", "Females")
df
## Class Males Females
## 1 Apartment 233 208
## 2 Dorm 159 138
## 3 With Parent(s) 102 280
## 4 Sorority/Fraternity House 220 265
## 5 Other 250 146
total<-sum(df$Males) + sum(df$Females)
round(sum(df$Males) / total, 4)
## [1] 0.4818
#6. Three cards are drawn with replacement from a standard deck. What is the probability that the first card will be a club, the second card will be a black card, and the third card will be a face card? Write your answer as a fraction or a decimal number rounded to four decimal places.
d1<-13/52
d2<-26/52
d3<-12/52
round(d1*d2*d3,4)
## [1] 0.0288
#7. Two cards are drawn without replacement from a standard deck of 52 playing cards. What is the probability of choosing a spade for the second card drawn, if the first card, drawn without replacement, was a heart? Write your answer as a fraction or a decimal number rounded to four decimal places.
#Find the value of p(B|A)
pA<-13/52
pB<-13/51
pAandB<-pA*pB
round(pAandB/pA, 4)
## [1] 0.2549
#8. Two cards are drawn without replacement from a standard deck of 52 playing cards. What is the probability of choosing a heart and then, without replacement, a red card? Write your answer as a fraction or a decimal number rounded to four decimal places.
#Conditional probability
pA<-15/52
pB<-25/51
round(pA*pB, 4)
## [1] 0.1414
#9. There are 85 students in a basic math class. The instructor must choose two students at random.
m <- c(12, 19, 12, 7)
f <- c(12, 15, 4, 4)
grade <- c("Freshmen", "Sophomores", "Juniors", "Seniors")
df <- data.frame(grade, m, f)
names(df) <- c("Level","Males", "Females")
df
## Level Males Females
## 1 Freshmen 12 12
## 2 Sophomores 19 15
## 3 Juniors 12 4
## 4 Seniors 7 4
p1 <- 4/85
p2 <- 12/84
p3 <- round(p1 * p2, 4)
p3
## [1] 0.0067
#10. Out of 300 applicants for a job, 141 are male and 52 are male and have a graduate degree.
#Step1
P_m <- 141/300
P_mgrad<- 52/300
round(P_mgrad / P_m, 4)
## [1] 0.3688
#Step2
P_d <- 102/300
round(P_mgrad / P_d, 4)
## [1] 0.5098
#11. A value meal package at Ron's Subs consists of a drink, a sandwich, and a bag of chips. There are 6 types of drinks to choose from, 5 types of sandwiches, and 3 types of chips. How many different value meal packages are possible?
d<-6
s<-5
c<-3
d*s*c
## [1] 90
#12. A doctor visits her patients during morning rounds. In how many ways can the doctor visit 5 patients during the morning rounds?
x<-1
y<-5
for(i in 1:y)
{
x <- x * i
print(x)
}
## [1] 1
## [1] 2
## [1] 6
## [1] 24
## [1] 120
#13. A coordinator will select 5 songs from a list of 8 songs to compose an event's musical entertainment lineup. How many different lineups are possible?
perm<-function(n,r)
{
factorial(n)/factorial(n-r)
}
perm(8,5)
## [1] 6720
#14. A person rolls a standard six-sided die 9 times. In how many ways can he get 3 fours, 5 sixes and 1 two?
factorial(9)/(factorial(3)*factorial(5)*factorial(1))
## [1] 504
#15. How many ways can Rudy choose 6 pizza toppings from a menu of 14 toppings if each topping can only be chosen once?
comb<-function(n,r){
factorial(n)/factorial(n-r)*factorial(r)
}
comb(14,6)
## [1] 1556755200
#16. 3 cards are drawn from a standard deck of 52 playing cards. How many different 3-card hands are possible if the drawing is done without replacement?
comb(52,3)
## [1] 795600
#17. You are ordering a new home theater system that consists of a TV, surround sound system, and DVD player. You can choose from 12 different TVs, 9 types of surround sound systems, and 5 types of DVD players. How many different home theater systems can you build?
d<-12*9*5
d
## [1] 540
#18. You need to have a password with 5 letters followed by 3 odd digits between 0 - 9 inclusively. If the characters and digits cannot be used more than once, how many choices do you have for your password?
perm(26,5)*perm(10,3)
## [1] 5683392000
#19. Evaluate the following expression. _9 P_4
perm(9,4)
## [1] 3024
#20. Evaluate the following expression. _11 C_8
comb(11,8)
## [1] 268240896000
#21. Evaluate the following expression. ( _12 P_8)/( _12 C_4 )
perm(12,8)/comb(12,4)
## [1] 70
#22. The newly elected president needs to decide the remaining 7 spots available in the cabinet he/she is appointing. If there are 13 eligible candidates for these positions (where rank matters), how many different ways can the members of the cabinet be appointed?
perm(13,7)
## [1] 8648640
#23. In how many ways can the letters in the word 'Population' be arranged?
factorial(10)/(factorial(2)*factorial(2))
## [1] 907200
#24. Consider the following data:
x <- c(5, 6, 7, 8, 9)
px <- c(0.1, 0.2, 0.3, 0.2, 0.2)
cdf<-data.frame(x,px)
cdf
## x px
## 1 5 0.1
## 2 6 0.2
## 3 7 0.3
## 4 8 0.2
## 5 9 0.2
#Step 1 Expected Value
expval<-sum(cdf$x+cdf$px)
expval
## [1] 36
#Step 2 Variance
Var<-sum((cdf$x - expval)^2 * cdf$px)
Var
## [1] 831
#Step 3 Standard Deviation.
sd<-sqrt(Var)
sd
## [1] 28.82707
#Step 4 p(x>=9)
with(cdf,sum(px[x>=9]))
## [1] 0.2
#Step 5p(x<=7)
with(cdf,sum(px[x<=7]))
## [1] 0.6
#25 Suppose a basketball player has made 188 out of 376 free throws. If the player makes the next 3 free throws, I will pay you $23. Otherwise you pay me $4.
px<-188/376
#Step 1 Expected Value of prop
expval<-round(23*(px^3) - 4*(1-px^3),2)
expval
## [1] -0.62
#Step 2 994 games
expval*994
## [1] -616.28
#26. Flip a coin 11 times. If you get 8 tails or less, I will pay you $1. Otherwise you pay me $7.
#Step 1 expected value. Using binomial
pw<-pbinom(8,size=11, prob=0.5)
pw
## [1] 0.9672852
expval<-round(1*pw - 7*(1-pw),2)
expval
## [1] 0.74
#Step 2 615 times
expval * 615
## [1] 455.1
#27. If you draw two clubs on two consecutive draws from a standard deck of cards you win $583. Otherwise you pay me $35. (Cards drawn without replacement.)
#Step 1: Expected Value
firstdrw<-13/52
secndrw<-12/51
p<-firstdrw*secndrw
expval<-round(p*583 - 35*(1-p),2)
expval
## [1] 1.35
#Step 2: 632 games
expval*632
## [1] 853.2
#28. A quality control inspector has drawn a sample of 10 light bulbs from a recent production lot. If the number of defective bulbs is 2 or less, the lot passes inspection. Suppose 30% of the bulbs in the lot are defective. What is the probability that the lot will pass inspection? (Round your answer to 3 decimal places)
#p(x>=2)
round(pbinom(2,10,.3),3)
## [1] 0.383
#OR
round(dbinom(0, 10, .3) + dbinom(1, 10, .3) + dbinom(2, 10, .3), 3)
## [1] 0.383
#29. A quality control inspector has drawn a sample of 5 light bulbs from a recent production lot. Suppose that 30% of the bulbs in the lot are defective. What is the expected value of the number of defective bulbs in the sample? Do not round your answer.
expval<-5 * .3
expval
## [1] 1.5
#30. The auto parts department of an automotive dealership sends out a mean of 5.5 special orders daily. What is the probability that, for any day, the number of special orders sent out will be more than 5? (Round your answer to 4 decimal places)
#p(x>5) Using Poisson dist
round(ppois(5,5.5, lower=FALSE),4)
## [1] 0.4711
#31. At the Fidelity Credit Union, a mean of 5.7 customers arrive hourly at the drive-through window. What is the probability that, in any hour, more than 4 customers will arrive? (Round your answer to 4 decimal places)
#p(x>4) poisson dist
round(ppois(4,5.7,lower=FALSE),4)
## [1] 0.6728
#32. The computer that controls a bank's automatic teller machine crashes a mean of 0.4 times per day. What is the probability that, in any 7-day week, the computer will crash no more than 1 time? (Round your answer to 4 decimal places)
#p(x<=1)
u<- 0.4 * 7 #Probaility for a week
round(ppois(1,u),4)
## [1] 0.2311
#34. A town recently dismissed 8 employees in order to meet their new budget reductions. The town had 6 employees over 50 years of age and 19 under 50. If the dismissed employees were selected at random, what is the probability that more than 1 employee was over 50? Write your answer as a fraction or a decimal number rounded to three decimal places.
round(phyper(1,m=6,n=19,8,lower.tail = FALSE),3)
## [1] 0.651
#35.Unknown to a medical researcher, 10 out of 25 patients have a heart problem that will result in death if they receive the test drug. Eight patients are randomly selected to receive the drug and the rest receive a placebo. What is the probability that less than 7 patients will die? Write your answer as a fraction or a decimal number rounded to three decimal places.
round(phyper(6,m=10,n=15,8),3)
## [1] 0.998