For those who were'nt able to come to class on Wednesday here it is. Mr. K start off our new topic,PROBABILITY. We did this little activity where we have to toss 3 coins (3 pennies) 30 times and we have to write how many times we got, exactly, 2 heads. One person got 16, highest, one person got 5, lowest, and 4 poeple got 10 which is the most. We haven't write anything that much, but here are some.
This is the theory that we came up with:
h = heads
t = tails
h t h t
h t h t h t h t
hhh hht hth htt thh tht tth ttt
3 out 8 chances that, exactly, 2 heads and 1 tail will show up. This is how you set it up:
Formula for PROBABILITY:
3 = Is the chance of getting exactly 2 heads and 1 tail
8 = Is the total # of event that can happen
P(e) = The number (#) of favourable outcome/The total (#) of outcome
P(e) = 3/8
0.375 = 3/8
Probability can be written in:
Decimal = .375
Fraction = 3/8
Ratio = 3/8
I hope you get something from this!
for next scribe Rein
[city b and city c] = 2 stopovers = k
and [city d] = destination = +1
C S R
C [ 65% 31% 4% ]
S [ 18% 70% 12% ]
R [ 7% 8% 75% ]
The number of people in each category in a particular year can be represented as 3 x 1. Matrix A= [c]
a) Formulate equations to represent the number of people the following year.
b) Summarise these relationships into a 3 x 3 transition matrix B
Enter the matrix B into your graphing calculator. To use the matrices to model the population change, you need to know what the initial population is. Suppose that c=250 000, y=150 000, a=500 000. Enter this information into the calculator as Matrix A, a 3x1 *column matrix.
*We're used to row matrices, keep that in mind while doing the multiplication (how we multiply)
With the above assumptions, the calculator will allow you to predict the next year's population as BA. It is best to start by evaluating BA; then find Mat B x Mat Ans.
c) What is the population predicted to be after one year? Describe the changes.
d) Repeat the command Mat B x Mat Ans and record the population for the next 14 years.
e) Comment on the distribution of the population over the 15 years.
f) Use a spreedsheet to table each category of the population over 50 years.
g) Graph the categories over this time.
h) Comment on the result and predict the future for the community.
TASK B 'Population Variations'
Investigate the long-term effect of the adult death rate rising from 2% per year to 4% per year.
Wow.. I wish I could say that I was almost done explaining what we did in class, but the fun's just beginning.
We had the pre-test. Then we got into our groups to correct. I suppose I could post the answers.. :P
1) Three stores, A, B, and C, were surveyed to determine costs of the same weight and brand of bread, flour, and sugar. The amounts to be purchased and the prices charged are shown in the matrices below. If all three items are purchased at the same store, what is the lowest total price?
bread flour sugar = [3 1 2]
Prices A B C
Bread [1.09 1.15 1.03 ]
Flour [6.15 5.95 6.45 ]
Sugar [4.32 4.30 4.10]
a) $11.40 b) 11.58 c) $17.74 d) $18.00 -- The answer is c. Just multiply Amounts matrix x Prices matrix.
2) A network matrix, N, shows direct routes between cities. Which matrix calculation determines the number of routes with exactly k stopovers between two cities in the network?
a) Nk -1 b) Nk c) Nk +1 d) Nk +2 -- The answer is C. I think a lot of people in the class got this one wrong. I honestly can't remember what the explanation was, my bad.
3) Determine (3Q)2 =
[1 2] All it is, is using scalar multiplication 3 x the matrix, then squaring it. The result is
4) Solve for x:
[1 0] [x ]
[x 6] [0 3][-2] = 0 . First, do x times 1, then x times 0. That makes x. Then it's 6 times o, and 6 times 3. That makes 18. Hence, [x 18]. x times x makes x2. 18 x -2 makes -36. Now, because we have to make this equal zero, x can be -6 or 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?
r l r l r l
[.70 .30] x r [.60 .40] = [.474 .526]
l [.18 .82]
This means that 47.4% is the next market share for regular powder, and 52.6 is the market share for large powder.
b) In the long term, what is the market share for each size of this washing powder?
For this, just do the same thing, except to the power of 50. :)
Wow, so close to being done.. yet so far. :P
The final thing we did (that I remember) was a group assignment regarding probability (next unit we're doing, by the way). Our test on matrices I believe is on Friday, or Wednesday. I can't remember if it's been settled ( sorry :( ).
Anyhow, the problem is as follows:
As we all know by now, our beloved Manitoba Moose are a team that is prong 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 and it seems like they will never win another game! Being the statistical sport that hockey is, Joe scoreboard (very lame :P) 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.
1. Write a transition matrix describing the situation above.
T = W [.70 .30] The sum of each row must equal 1 (100%).
L [.40 .60]
2) Suppose that the moose won their first game of the season. What would the initial distribution (initial state) matrix?
D0[1 0] 1 represents that they won, while 0 represents that there is no chance they lost, because well, they won.
3) Given the conditions from 1 and 2, find the Moose's probable winning percentage after 1,2, 3, 4, 5, and 10 games. Mulitplythe matrix in question 2 by the one in number 1.
For example, Do10 = D0T10 ... What does D10 represent? ... For this one, I believe it means the number of games.
1) [.7 .3]
2) [.61 .39]
3) [.583 .417]
4) [.5759 .4251]
5) [.5725 .4275]
10) [.5714] [.4286]
Having played 10 games, could the Moose actually have a record represented by this posibility?
The answer is that they can't, because after 10 games you can't have won 57% of them.
4) Now, recalculate example 3 with the condition that the Moose lost their first game. Complete the chart below and compare the results. *To get the probabilities from having lost the first game, just change the state matrix from [1 0] to [0 1]
Winning percentage having won or lost first game.
1 game .70 .4
2 games .61 .52
3 games .583 .556
4 games .5749 .5568
5 games .5725 .5700
10 games .5714 .5714
5) Notice that there is very little difference between the results after 5 games and after 10 games. What do these percentages represent?
These percentages represent the matrix stabilizing, meaning they can expect to win 57% of all the games they play.
7) The Moose play an 80 game regular season. Predict their final won-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?
The records are the same, [.5714 .4286] .... This means that regardless of how their season begins, they may win 57% of all games they play. Therefore, the initial state doesn't really mean anything over time.
I am very sorry that it took me so long to post this (I also apologize for the length!). I hope I explained everything clearly and properly. Monday's scribe will be Mohamed.
We've learned about two different kinds of mutliplication that can be done with matricies; scalar multiplication and matrix multiplication. There's also ordinary multiplication, such as 2 x 3 = 6. Blog a brief paragraph identifying ways in which these three types of multiplication are similar. Blog a second paragraph outlining the ways in which they are different. Also, give us an example of two 2x2 matricies, A and B, that DO commute. i.e. AB = BA.
Written as: r x c Expressed as: "r by c"
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