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.
library(MASS)
print(paste0("The probability of reaching into the box and randomly drawing the chip numbered 505 is ", fractions(1/540), "."))## [1] "The probability of reaching into the box and randomly drawing the chip numbered 505 is 1/540."extraToppings <- c('A', 'C')
meats <- c('E', 'T')
dressings <- c('F', 'V')
for (t in extraToppings){
for (m in meats){
for (d in dressings){
print(paste(t,m,d, sep=','))
}
}
}## [1] "A,E,F"
## [1] "A,E,V"
## [1] "A,T,F"
## [1] "A,T,V"
## [1] "C,E,F"
## [1] "C,E,V"
## [1] "C,T,F"
## [1] "C,T,V"# 52 playing cards
# 13 of them are hearts
# 10 of hearts are not a face card
print(paste0("The probability that the card will be a heart and not a face card is ", fractions(10/52), "."))## [1] "The probability that the card will be a heart and not a face card is 5/26."die1 <- 1:6
die2 <- 1:6
moreThan6 <- 0
lessThan6 <- 0
for (d1 in die1){
for (d2 in die2){
if(d1 + d2 < 6){
lessThan6 <- lessThan6 + 1
}
moreThan6 <- moreThan6 + 1
}
}
print(paste0("The probability of rolling a sum less than 6 is ", fractions(lessThan6/moreThan6), "."))## [1] "The probability of rolling a sum less than 6 is 5/18."| Gender and Residence of Customers | Males | Females |
|---|---|---|
| Apartment | 233 | 208 |
| Dorm | 159 | 138 |
| With Parents(s) | 102 | 280 |
| Sorority/Fraternity House | 220 | 265 |
| Other | 250 | 146 |
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)
females <- c(208, 138, 280, 265, 146)
total <- sum(males + females)
print(paste0("The probability that a customer is male is ", fractions(sum(males)/total), "."))## [1] "The probability that a customer is male is 964/2001."print(paste0("The probability of a club card is ", fractions(13/52), "."))## [1] "The probability of a club card is 1/4."print(paste0("The probability of a black card is ", fractions(26/52), "."))## [1] "The probability of a black card is 1/2."print(paste0("The probability of a face card is ", fractions(12/52), "."))## [1] "The probability of a face card is 3/13."heart <- 13/52
club <- 13/51
print(paste0("The probability of choosing a spade for the second card drawn, if the first card, drawn without replacement, was a heart is ", fractions((heart*club)/heart), "%."))## [1] "The probability of choosing a spade for the second card drawn, if the first card, drawn without replacement, was a heart is 13/51%."heart <- 13/52
red <- 25/51
print(paste0("The probability of choosing a heart and then, without replacement, a red card is ", fractions(heart*red), "."))## [1] "The probability of choosing a heart and then, without replacement, a red card is 25/204."| Students in a Basic Math Class | Males | Females |
|---|---|---|
| Freshmen | 12 | 12 |
| Sophomores | 19 | 15 |
| Juniors | 12 | 4 |
| Seniors | 7 | 4 |
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.
jFemale <- 4/85
fMale <- 12/85
print(paste0("The probability that a junior female and then a freshmen male are chosen at random is ", fractions(jFemale*fMale), "."))## [1] "The probability that a junior female and then a freshmen male are chosen at random is 48/7225."male <- 141/300
gdMale <- 52/300
gd <- 102/300
print(paste0("The probability that a randomly chosen applicant has a graduate degree, given that they are male is ", fractions(gdMale/male), "."))## [1] "The probability that a randomly chosen applicant has a graduate degree, given that they are male is 52/141."print(paste0("The probability that a randomly chosen applicant is male, given that the applicant has a graduate degree is ", fractions(gdMale/gd), "."))## [1] "The probability that a randomly chosen applicant is male, given that the applicant has a graduate degree is 26/51."drink <- 6
sandwich <- 5
chips <- 3
print(paste(drink*sandwich*chips, "different value meal packages are possible."))## [1] "90 different value meal packages are possible."print(paste("In", factorial(5), "many ways the doctor can visit 5 patients during the morning rounds."))## [1] "In 120 many ways the doctor can visit 5 patients during the morning rounds."Permutations <- function(n,r){
factorial(n)/factorial(n-r)
}
print(paste(Permutations(8,5), "different lineups are possible."))## [1] "6720 different lineups are possible."fours <- 3
sixes <- 5
two <- 1
rolls <- 9
print(paste(factorial(rolls)/((factorial(fours)*factorial(sixes)*factorial(two))), "ways."))## [1] "504 ways."Combinations <- function(n,r){
factorial(n)/(factorial(n-r)*factorial(r))
}
paste(paste("Rudy can choose 6 pizza toppings from a menu of 14 toppings in", Combinations(14,6), "ways."))## [1] "Rudy can choose 6 pizza toppings from a menu of 14 toppings in 3003 ways."print(paste(fractions(Combinations(52,3)), "different 3-card hands are possible if the drawing is done without replacement."))## [1] "22100 different 3-card hands are possible if the drawing is done without replacement."tv <- 12
sound <- 9
dvd <- 5
print(paste(tv*sound*dvd, "different home theatre systems can be build."))## [1] "540 different home theatre systems can be build."print(paste(Permutations(26,5)*Permutations(5,3), "many ways to create the password."))## [1] "473616000 many ways to create the password."Permutations(9,4)## [1] 3024Combinations(11,8)## [1] 165Permutations(12,8)/Combinations(12,4)## [1] 40320print(paste(Permutations(13,7), "ways the members of the cabiner can be appointed."))## [1] "8648640 ways the members of the cabiner can be appointed."p <- 2
o <- 2
total <-10
print(paste(factorial(10)/(factorial(2)*factorial(2)), "ways the letters in the word 'Population' can be arranged." ))## [1] "907200 ways the letters in the word 'Population' can be arranged."| \(x\) | \(p(x)\) |
|---|---|
| 5 | 0.1 |
| 6 | 0.2 |
| 7 | 0.3 |
| 8 | 0.2 |
| 9 | 0.2 |
x <- c(5,6,7,8,9)
px <- c(0.1,0.2,0.3,0.2,0.2)
xpx <- data.frame(x, px)
ev <- sum(xpx$x * xpx$px)
var <- sum((xpx$x - ev)^2 * xpx$px)
sd <- sqrt(var)
ninePlus <- sum((x>=9) * px)
sevenMinus <- sum((x<9) * px)
print(paste0("The expected value is ", round(ev, 1), "."))## [1] "The expected value is 7.2."print(paste0("The variance is ", round(var, 1)))## [1] "The variance is 1.6"print(paste0("The standard deviation is ", round(sd, 1), "."))## [1] "The standard deviation is 1.2."print(paste0("The value of P(X=>9) is ", round(ninePlus, 1), "."))## [1] "The value of P(X=>9) is 0.2."print(paste0("The value of P(X<7) is ", round(sevenMinus, 1), "."))## [1] "The value of P(X<7) is 0.8."throwIn <- 188
total <- 376
shot1 <- throwIn/total
shot2 <- shot1^2
shot3 <- shot1^3
throwOut <- 1 - shot3
gain <- shot3 * 23
loss <- throwOut * -4
print(paste0("The expected value of the proposition is ", round(gain+loss, 2), "."))## [1] "The expected value of the proposition is -0.62."print(paste0("I would lose $", round(gain + loss,2) * 994, ", if this game was played 994 times."))## [1] "I would lose $-616.28, if this game was played 994 times."gain <- pbinom(8, size=11, prob=.5)
print(paste0("The expected value of the proposition is ", round(gain*1 + (1-gain)*-7, 2), "."))## [1] "The expected value of the proposition is 0.74."print(paste0("If I played this game 615 times, I would win $", round((gain*1 + (1-gain)*-7)*615, 2)))## [1] "If I played this game 615 times, I would win $454.04"club1 <- 13/52
club2 <- 12/51
bothClub <- club1 * club2
noClub <- 1 - bothClub
ev <- bothClub*583 + noClub*-35
print(paste0("The expected value of the proposition is ", round(ev, 2), "."))## [1] "The expected value of the proposition is 1.35."print(paste0("If I played this game 632 times, I would win $", round(ev*632, 2), "."))## [1] "If I played this game 632 times, I would win $855.06."pass <- pbinom(2, size = 10, prob = .3)
print(paste0("The probability that the lot will pass inspection is ", round(pass, 3), "%." ))## [1] "The probability that the lot will pass inspection is 0.383%."print(paste0("The expected value of the number of defective bulbs in the sample is ", 5*.3, "."))## [1] "The expected value of the number of defective bulbs in the sample is 1.5."print(paste0("The probability that, for any day, the number of special orders sent out will be more than 5 is ", round(ppois(5, 5.5, lower.tail = FALSE),4), "%."))## [1] "The probability that, for any day, the number of special orders sent out will be more than 5 is 0.4711%."print(paste0("The probability that, in any hour, more than 4 customers will arrive is ", round(ppois(4, 5.7, lower.tail = FALSE), 4), "%."))## [1] "The probability that, in any hour, more than 4 customers will arrive is 0.6728%."print(paste0("The probability that, in any 7-day week, the computer will crash no more than 1 time is ", round(ppois(1, 0.4*7, lower.tail = TRUE), 4), "%."))## [1] "The probability that, in any 7-day week, the computer will crash no more than 1 time is 0.2311%."dismissed <- 8
hired <- 6
age1 <- 19
age2 <- 50
hyperDist <- phyper(1, m = hired, n = age1, dismissed, lower.tail = FALSE)
print(paste0("The probability that more than 1 employee was over 50 is ", round((hyperDist), 4), "%."))## [1] "The probability that more than 1 employee was over 50 is 0.6506%."heartProb <- 10
drugTest <- 8
test <- 15
patients <- 25
hyperDist <- phyper(6, m = heartProb, n = test, drugTest)
print(paste0("The probability that less than 7 patients will die is ", round((hyperDist), 3), "%."))## [1] "The probability that less than 7 patients will die is 0.998%."