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.
prob <- 1/540
fractions(prob)## [1] 1/540Write 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.)
The sample space is as follows: {AEF,AEV, ATF, ATV, CEF, CEV, CTF,CTV}
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.
Let H represent the event that the card is a heart Let NF represent the event that the card is not a face card
Notes:
In a standard 52 card deck there are 12 face cards link Therefore, there are 52-12 = 40 cards that are not face cards
In a standard 52 card deck there are 13 hearts.
Therefore: P(H and NF) = P(H) * P(NF)
probH <- 13/52
probNF <- 40/52
Ans3 <- probH * probNF
fractions(Ans3)## [1] 5/26A 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.
36 total possibilities Less than 6: {4/1,3/2,3/1,2/3,2/2,2/1,1/4,1/3,1/2,1/1} - 10 possibities
fractions(10/36)## [1] 5/18A 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.
Gender and Residence of Customers
Males Females
Apartment 233 208
Dorm 159 138
With Parent(s) 102 280
Sorority/Fraternity House 220 265
Other 250 146What is the probability that a customer is male? Write your answer as a fraction or a decimal number rounded to four decimal places.
Total <- 233+159+102+220+250+208+138+280+265+146
Male <- 233+159+102+220+250
ProbMale <- Male/Total
fractions(ProbMale)## [1] 964/2001Three 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.
Let C be the event that the card is a club Let B be the event that the card is black Let F be the event that the card is a face card
probC <- 13/52
probB <- 26/52
probF <- 12/52
probCBF <- probC * probB * probF
fractions(probCBF)## [1] 3/104Two 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.
Let S represent the event that a spade is drawn on second draw without replacement Let H represent the event that a heart is drawn
probH <- 13/52
probS <- 13/51
probSgivenH <- probH * probS
fractions(probSgivenH)## [1] 13/204Two 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.
Let H represent the event that a heart is chosen Let R represent the event that a red card is chosen
PHandR <- 13/52 * 25/51
fractions(PHandR)## [1] 25/204There are 85 students in a basic math class. The instructor must choose two students at random.
Students in a Basic Math Class
Males Females
Freshmen 12 12
Sophomores 19 15
Juniors 12 4
Seniors 7 4What is the probability that a junior female and then a freshmen male are chosen at random? Write your answer as a fraction or a decimal number rounded to four decimal places.
Let JF represent the event that a Junior Female is chosen Let FM represent the event that a Freshman Male is chosen
Total <- 12+19+12+7+12+15+4+4
PJFthenFM <- 4/Total * 12/(Total-1)
fractions(PJFthenFM)## [1] 4/595Out of 300 applicants for a job, 141 are male and 52 are male and have a graduate degree.
Step 1. What is the probability that a randomly chosen applicant has a graduate degree, given that they are male? Enter your answer as a fraction or a decimal rounded to four decimal places.
Step 2. If 102 of the applicants have graduate degrees, what is the probability that a randomly chosen applicant is male, given that the applicant has a graduate degree? Enter your answer as a fraction or a decimal rounded to four decimal places.
Let M represent the event that the applicant is male Let G represent the event that the applicant has a graduate degree
ProbM <- 141/300
ProbGandM <- 52/300
PGgivenM <- ProbGandM/ProbM
fractions(PGgivenM)## [1] 52/141ProbG <- 102/300
PMgivenG <- ProbGandM/ProbG
fractions(PMgivenG)## [1] 26/51A 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?
Ans <- 6 * 5 * 3
Ans## [1] 90A doctor visits her patients during morning rounds. In how many ways can the doctor visit 5 patients during the morning rounds?
Ans <- factorial(5)
Ans## [1] 120A 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?
Assume order of lineup is important.
nrow(permutations(8,5))## [1] 6720A person rolls a standard six-sided die 9 times. In how many ways can he get 3 fours, 5 sixes and 1 two?
Ans <- 3 * 5 * 1
Ans## [1] 15How many ways can Rudy choose 6 pizza toppings from a menu of 14 toppings if each topping can only be chosen once?
nrow(combinations(14,6))## [1] 30033 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?
nrow(combinations(52,3))## [1] 22100You 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?
Ans <- 12 * 9 * 5
Ans## [1] 540You 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?
Assume letters must be lowercase only Letters - there are 26 letters Digits - there are 5 odd digits between 0 and 9 (1,3,5,7,9)
Letters <- nrow(permutations(26,5))
Digits <- nrow(permutations(5,3))
Password <- Letters * Digits
Password## [1] 473616000Evaluate the following expression.9P4
Ans <- nrow(permutations(9,4))
Ans## [1] 3024Evaluate the following expression. 11C8
Ans <- nrow(combinations(11,8))
Ans## [1] 165Evaluate the following expression. (12P8)/( 12C4)
Ans <- (nrow(permutations(12,8))/nrow(combinations(12,4)))
Ans## [1] 40320The 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?
nrow(permutations(13,7))## [1] 8648640In how many ways can the letters in the word ‘Population’ be arranged?
Ans <- factorial(10)
Ans## [1] 3628800Consider the following data:
x 5 6 7 8 9 p(x) 0.1 0.2 0.3 0.2 0.2
Step 1. Find the expected value E( X ). Round your answer to one decimal place.
Step 2. Find the variance. Round your answer to one decimal place. Step 3. Find the standard deviation. Round your answer to one decimal place. Step 4. Find the value of P(X >= 9). Round your answer to one decimal place. Step 5. Find the value of P(X <= 7). Round your answer to one decimal place.
x <- c(5,6,7,8,9)
p <- c(0.1,0.2,0.3,0.2,0.2)
step1 <- round(sum(x*p),1)
step1## [1] 7.2step2 <- round(((sum(x^2 * p)) - step1^2),1)
step2## [1] 1.6step3 <- round(sqrt(step2),1)
step3## [1] 1.3step4 <- 0.2
step4## [1] 0.2step5 <- 0.1+0.2+0.3
step5## [1] 0.6Suppose 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.
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.)
Let X represent the event that the number of free throws made by the basketball player : x ~ B(3,188/376)
Step1: Expected Value
Ans <- 3 * 188/376
round(Ans,2)## [1] 1.5Step2: Expected Wins or Losses:
probLoss <- pbinom(2, size=3, prob=188/376)
probWin <- dbinom(3, size=3, prob=188/376)
Losses <- -4 * probLoss*994
Wins <- 23*probWin * 994
Winnings <- Losses + Wins
Winnings## [1] -621.25Flip a coin 11 times. If you get 8 tails or less, I will pay you $1. Otherwise you pay me $7.Step 1. Find the expected value of the proposition. Round your answer to two decimal places.
Step 2. If you played this game 615 times how much would you expect to win or lose? (Losses must be entered as negative.)
Let X represent the number of tail tossed : x ~ B(11,1/2)
Step1: Expected Value
Ans <- 11 * 1/2
round(Ans,2)## [1] 5.5Step2: Expected Wins or Losses:
probLoss <- pbinom(8, size=11, prob=1/2, lower.tail = FALSE)
probWin <- dbinom(8, size=11, prob=1/2)
Losses <- -7 * probLoss*615
Wins <- 1*probWin * 615
Winnings <- Losses + Wins
Winnings## [1] -91.28906If 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. Find the expected value of the proposition. Round your answer to two decimal places.
Step 2. If you played this game 632 times how much would you expect to win or lose? (Losses must be entered as negative.)
Since cards are drawn without replacement the distribution is hypergeometric
Expected Value:
n <- 2
k <- 13
N <- 52
Evalue <- (n*k)/N
round(Evalue,2)## [1] 0.5Earnings
pWin <- dhyper(2,13,39,2)
pLoss <- 1 - pWin
Winnigs <- 583*pWin*615 + (-35)*pLoss*615
Winnings## [1] -91.28906A 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)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.
Expected Value:
Ans <- 5 * 0.3
Ans## [1] 1.5The 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)
round(ppois(5, lambda = 5.5, lower.tail = FALSE),4)## [1] 0.4711At 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)
round(ppois(4, lambda = 5.7, lower.tail = FALSE),4)## [1] 0.6728The 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)round(ppois(1, lambda = 2.8),4)## [1] 0.2311A 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.
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.