Pre-test, review of transition matrices, and homework.

1. Matrices Pre-Test
and Workshop ...
Studying for Discrete Math

6. (5) A washing powder is sold in regular and large boxes. Market research shows
that 40% of the buyers of the regular size switch to the large size for their next
purchase, and 18% of the buyers of the large size switch to the regular size for their
next purchase. The original market share was 70% for the regular size and 30% for
the large size.
(a) What is the market share for each size in the next round of purchases?
(Show the transition matrix used.) (3 marks)
(b) In the long term, what is the market share for each size of this washing
powder? (2 marks)

7. Manitoba Moose: Winning Streaks and Losing Streaks
As we all know by now, our beloved Manitoba Moose are a team that is
prone to streaky play. In other words, when they are winning, it seems like
they will never lose again. On the other hand, there are times when they
are losing that it seems like they will never win another game! Being the
statistical sport that hockey is, Joe Scoreboard has made the following
observations concerning the Moose. That is, when the Moose win a game,
there is a 70% chance that they will win their next game. However, when
the Moose lose a game, they have only a 40% chance of winning their next
game. Use this data to answer the following questions.

8. Having played 10 games, could the Moose actually have a record represented by
this probability ? Why or why not ?

10. 5. Notice that there is very little difference between the results after 5 games and after
10 games. What do these percentages represent?
6. After how many games does the state matrix stabilize (become constant)
to 4 decimal places? Use matrices to ilustrate this stable state.
7. The Moose play an 80 game regular season. Predict their final win - loss
record assuming that they win their first game and also if they lose their
first game. How do the records compare? In this example, how important is
the initial state?

11. HOMEWORK
Wickers Furniture has two retail stores, A and B. Their inventory for the
number of sofas, recliners, dressers, and dinettes is given in the table below.
sofas recliners dressers dinettes
Store A 7 12 10 19
Store B 15 6 8 9
Sofas cost $800 each, recliners $525 each, dressers $650 each, and
dinettes $750 each.
(a) Write a 2 x 4 matrix for the inventory and name it A.
(b) Write a 4 x 1 column matrix that shows the prices, and name it B.
(c) Calculate AB to determine the value of the inventory at each store.
Label the new matrix C, and save it.
(d) Write a 4 x 4 square matrix where the elements of the main
diagonal are the prices, and all other elements are zeros. Label this
matrix D. Find AD. What do the values in matrix AD represent?

12. HOMEWORK
A large grain cleaning plant has five centres A can move grain to B and D
of operation identified as A, B, C, D, and E. B can move grain to A and C
These sites are connected with conveyor belts C can move grain to D
to move grain as follows: D can move grain to E
E can move grain to A and C
(a) Create a square matrix A to (b) Write matrix B to represent the
represent the direct routes between number of ways grain can be moved
sites. A '1' represents a direct route, between sites via one other site.
and a '0' represents no direct route.
(c) Can grain be moved from any site to any other site using no more than two
trips on the conveyor belt? If not, then indicate which sites cannot be
connected in this way. Explain how you determine this answer.