DSMath_Week2

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. \[\frac{1}{540}\]

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.)

toppings <- c("A", "C")
meats <- c("E", "T")
dressings <- c("F", "V")

perms <- expand.grid(toppings, meats, dressings)
paste(perms$Var1, perms$Var2, perms$Var3, collapse = ", ")

## [1] "A E F, C E F, A T F, C T F, A E V, C E V, A T V, 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. \[\frac{9}{52}\]
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.

ss <- expand.grid(1:6, 1:6)
ss$total = ss$Var1 + ss$Var2
lt6 <- filter(ss, total < 6)

round(count(lt6)/count(ss), 4)

##        n
## 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. What is the probability that a customer is male? Write your answer as a fraction or a decimal number rounded to four decimal places.

males <- c(233,159,102,220,250)
round(sum(males)/2001, 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.

pClub <- 13/52
pBlack <- 26/52
pFace <- 12/52

round(pClub * pBlack * pFace, 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.

round(13/51, 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.

heart <- 13/52
red <- 25/51

round(heart * red, 4)

## [1] 0.1225

9. There are 85 students in a basic math class. The instructor must choose two students at random. What 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.

jF <- 4/85
fM <- 12/84

round(jF * fM, 4)

## [1] 0.0067

10. Out of 300 applicants for a job, 141 are male and 52 are male and have a graduate degree.
    • 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.

    round(52/141, 4)

    ## [1] 0.3688

    • 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.

    round(52/102, 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?

6 * 5 * 3

## [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?

factorial(5)

## [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?

nPr(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?

15. How many ways can Rudy choose 6 pizza toppings from a menu of 14 toppings if each topping can only be chosen once?

nCr(14,  6)

## [1] 3003

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?

nCr(52, 3)

## [1] 22100

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?

prod(c(12, 9, 5))

## [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?

odds <- c(1, 3, 5, 7, 9)
nPr(26, 5) * nPr(length(odds), 3)

## [1] 473616000

19. Evaluate the following expression. \(_{9}P_{4}\)

nPr(9, 4)

## [1] 3024

20. Evaluate the following expression. \(_{11}C_{8}\)

nCr(11, 8)

## [1] 165

21. Evaluate the following expression. \(\frac{_{12}P_{8}}{_{12}C_{4}}\)

nPr(12, 8)/nCr(12, 4)

## [1] 40320

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?

nPr(13, 7)

## [1] 8648640

23. In how many ways can the letters in the word ‘Population’ be arranged?

word <- "population"
letterVec <- sapply(unlist(strsplit(word, "")), tolower)
nPr(length(letterVec), length(letterVec))

## [1] 3628800

24. Consider the following data:
    • Step 1. Find the expected value \(E(X)\). Round your answer to one decimal place.

    x <- 5:9
    px <- c(.1,.2,.3,.2,.2)
    xpx <- x * px
    distro <- cbind(x, px, xpx) %>%
      data.frame
    
    ex <- sum(distro$xpx)/sum(distro$px)
    round(ex, 1)

    ## [1] 7.2

    • Step 2. Find the variance. Round your answer to one decimal place.

    variance <- (sum((distro$x - ex)^2 * distro$px))/sum(distro$px)
    round(variance, 1)

    ## [1] 1.6

    • Step 3. Find the standard deviation. Round your answer to one decimal place.

    sd <- sqrt(variance)
    round(sd, 1)

    ## [1] 1.2

    • Step 4. Find the value of \(P(X >= 9)\). Round your answer to one decimal place.

    gt9 <- subset(distro$px, distro$x >= 9)
    round(sum(gt9), 1)

    ## [1] 0.2

    • Step 5. Find the value of \(P(X <= 7)\) Round your answer to one decimal place.

    lt7 <- subset(distro$px, distro$x <= 7)
    round(sum(lt7), 1)

    ## [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.
    • Step 1. Find the expected value of the proposition. Round your answer to two decimal places.

    ex <- dbinom(3, 3, 188/376) * 23 + (1 - dbinom(3, 3, 188/376)) * -4
    round(ex, 2)

    ## [1] -0.62

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

    round(ex * 994, 2)

    ## [1] -621.25

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. Find the expected value of the proposition. Round your answer to two decimal places.

    ex <- pbinom(8, 11, .5) * 1 + pbinom(8, 11, .5, lower.tail = F) * -7
    round(ex, 2)

    ## [1] 0.74

    • Step 2. If you played this game 615 times how much would you expect to win or lose? (Losses must be entered as negative.)

    round(ex * 615, 2)

    ## [1] 454.04

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. Find the expected value of the proposition. Round your answer to two decimal places.

    p <- dhyper(2, 13, 39, 2)
    ex <- p * 583 + (1 - p) * -35
    round(ex, 2)

    ## [1] 1.35

    • Step 2. If you played this game 632 times how much would you expect to win or lose? (Losses must be entered as negative.)

    round(ex * 632, 2)

    ## [1] 855.06

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)

ex <- pbinom(2, 10, .3)
round(ex, 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.

5 * .3

## [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)

ppois(5, 5.5, lower.tail = F) %>%
  round(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)

ppois(4, 5.7, lower.tail = F) %>%
  round(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)

ppois(1, 2.8) %>%
  round(4)

## [1] 0.2311

33. 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.

phyper(1, 6, 19, 8, lower.tail = F) %>%
  round(3)

## [1] 0.651

34. 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.

phyper(6, 10, 15, 8) %>%
  round(3)

## [1] 0.998