Statistics Help

Question

Some questions I could use help on

A director of a music video show wishes to play 4 rap videos and 3 R&B videos. He has 8 rap and 6 R&B videos to choose from. If we consider the order in which the videos are played, how many different line-ups can the show have? Note that no video can be used more than once.

On an assignment, students are expected to answer at least 2 questions from a 19 question set. But they can complete as many questions as they wish (for bonus!). How many possible ways can they complete the assignment? Note: Consider all questions do-able, and each student will complete at least the minimum number of questions.

A car is picked at random from the west campus parking lot. Let E be the event "the car is blue". Let F be the event "the car is a Toyota". Let G be the event "the car is a green Ford". Which of the following pairs of events are mutually exclusive? (i) E and G (ii) G and E ∪ F (iii) F′ ∪ G′ and E

Answer 1

You can select 4 of 8 rap videos and 3 of 6 R&B videos in C(8,4) x C(6,3) ways. As we are playing 7 videos, and order matters, there are 7! orders in which they may be played.

Thus, C(8,4) x C(6,3) x 7! = 70 * 20 * 5040 = 7056000

As there are 19 questions that can be answered or not answered, there are 2^19 possible ways to select them. However, as we must select at least two, we subtract the number of ways that you can select 0 (1) or1(as there are 19 questions, 19)

Thus, 2^19 - 19 - 1 = 2^19 - 20 = 524268

i) E, the car is blue , and G, the car is a green Ford. As being blue and green are mutually exclusive, and G is a subset of the set of green cars, E and G are mutually exclusive

ii) G and E U F If G is mutually exclusive of E and G is mutually exclusive of F, G is mutually exclusive of E and F. From i), we know that it is mutually exclusive of E. F is a car is a Toyota and G is a car being a green Ford. As being a Toyota and being a Ford are mutually exclusive, E and G are mutually exclusive.

Thus, G is mutually exclusive of E and F, so G is mutually exclusive of E U F

iii) We make the assumption here that E contains at least one member. Then, the complement of being a green Ford is not being green or not being a Ford. Then, being blue is a subset of not being green, so E is a subset of G', and E is a subset of F' U G'. Thus, E and F' U G' are not mutually exclusive.