DATA650_HW6.Rmd

1. A bag contains 5 green and 7 red jellybeans. How many ways can 5 jellybeans be withdrawn from the bag so that the number of green ones withdrawn will be less than 2?

There are two possible proportions that satisfy the less than 2 requirement: * 6 red jellybeans/7 total jellybeans * 7 red jellybeans/7 total jellybeans

one_green <- choose(5,1) * choose(7,4)        
zero_green <- choose(5,0) * choose(7,5)

print(one_green + zero_green)

## [1] 196

2. A certain congressional committee consists of 14 senators and 13 representatives. How many ways can a subcommittee of 5 be formed if at least 4 of the members must be representatives?

There are 2 possible proportions that satisfy the at least 4 representatives requirement. * 4 representatives/5 members * 5 representatives/5 members

four_reps<-choose(14,1) * choose(13,4)       
five_reps<-choose(14,0) * choose(13,5) 

print(four_reps + five_reps)

## [1] 11297

3. If a coin is tossed 5 times, and then a standard six-sided die is rolled 2 times, and finally a group of three cards are drawn from a standard deck of 52 cards without replacement, how many different outcomes are possible?

2^5 * 6^2 * 52 * 51 * 50

## [1] 152755200

4. 3 cards are drawn from a standard deck without replacement. What is the probability

that at least one of the cards drawn is a 3? Express your answer as a fraction or a decimal number rounded to four decimal places.

What is the chance that each card is not a 3 and find the compliment?
There are four 3s in a deck so the chance of getting a non-3 is 48/52 and it decreases incrementally for each additional card

round(1 - (48/52 * 47/51 * 46/50),4)

## [1] 0.2174

6. Lorenzo is picking out some movies to rent, and he is primarily interested in

documentaries and mysteries. He has narrowed down his selections to 17 documentaries and 14 mysteries.

all_combs <-choose(31 ,5)

print(all_combs)

## [1] 169911

How many different combinations of 5 movies can he rent if he wants at least one mystery? Take all the possible combinations and remove the combinations that are just from the documentaries?

just_docs <- choose(17,5)

print(all_combs - just_docs)

## [1] 163723

7. In choosing what music to play at a charity fund raising event, Cory needs to have an

equal number of symphonies from Brahms, Haydn, and Mendelssohn. If he is setting up a schedule of the 9 symphonies to be played, and he has 4 Brahms, 104 Haydn, and 17 Mendelssohn symphonies from which to choose, how many different schedules are possible? Express your answer in scientific notation rounding to the hundredths place.

brahms <- choose(4,3)
hayden <- choose(104,3)
mendel <- choose(17,3)

options(scipen = 1)
options(digits = 4)

print(brahms * hayden * mendel)

## [1] 495322880

7. An English teacher needs to pick 13 books to put on his reading list for the next school year, and he needs to plan the order in which they should be read. He has narrowed down his choices to 6 novels, 6 plays, 7 poetry books, and 5 nonfiction books.

If he wants to include no more than 4 nonfiction books, how many different reading schedules are possible? Express your answer in scientific notation rounding to the hundredths place.

find all combinations and subtract off the number of combinations where all 5 nonfiction are not optional

all <-choose(24,13)
not_optional<- choose(19, 8)

options(scipen = 1)
options(digits = 4)


print(all-not_optional)

## [1] 2420562

If he wants to include all 6 plays, how many different reading schedules are possible? Express your answer in scientific notation rounding to the hundredths place.

This is similar to the first part, just don’t include the plays

options(digits = 3, scipen = -3)
choose(18,7)

## [1] 31824

8. Zane is planting trees along his driveway, and he has 5 sycamores and 5 cypress trees to

plant in one row. What is the probability that he randomly plants the trees so that all 5 sycamores are next to each other and all 5 cypress trees are next to each other? Express your answer as a fraction or a decimal number rounded to four decimal places

It’s important to remember that all the sycamores can be first or the cypress can be first

probabiltiy <- (factorial(5)*factorial(5))/factorial(10)
probabiltiy <- probabiltiy  * 2

print(round(probabiltiy, 4))

## [1] 7.9e-03

9. If you draw a queen or lower from a standard deck of cards, I will pay you $4. If not, you

pay me $16. (Aces are considered the highest card in the deck.) Step 1. Find the expected value of the proposition. Round your answer to two decimal places. Losses must be expressed as negative values.

There are 8 cards were I lose and 44 cards where I win

win <- 44/52
lose <- 8/42

expected <- (4 * win) + (-16 * lose)

round(expected, 2)

## [1] 0.34

If you played this game 833 times how much would you expect to win or lose? Round your answer to two decimal places. Losses must be expressed as negative values.

round(833 * expected, 2)

## [1] 281