Tugasan 2

Part 1
(a) Ok, this is about history of probability and its applications. There are tonnes of internet web page discussing about this topic, so I think this shouldn't be any problem to you all, right?

(b) This is about theoretical and empirical probability. There are also tonnes of discussion in internet. So, just Google it. Basically, theoretical knowledge means knowledge that you obtain through your thinking, while empirical knowledge is knowledge obtain through your experience and experiments.

Part 2
(a) {1,2,3,4,5,6}
(b)

Chart

Table
Tree Diagram

Total Outcome
{ (1,1), (1,2), (1,3), (1,4), (1,5), (1,6)
(2,1), (2,2), (2,3), (2,4), (2,5), (2,6)
(3,1), (3,2), (3,3), (3,4), (3,5), (3,6)
(4,1), (4,2), (4,3), (4,4), (4,5), (4,6)
(5,1), (5,2), (5,3), (5,4), (5,5), (5,6)
(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

Part 3
(a)


(b)
A = { (1,1), (1,2), (1,3), (1,4), (1,5), (1,6)
(2,1), (2,2), (2,3), (2,4), (2,5), (2,6)
(3,1), (3,2), (3,3), (3,4), (3,5), (3,6)
(4,1), (4,2), (4,3), (4,4), (4,5), (4,6)
(5,1), (5,2), (5,3), (5,4), (5,5), (5,6)
(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

B = ø

P = Both number are prime
P = {(2,2), (2,3), (2,5), (3,3), (3,5), (5,3), (5,5)}



Q = Difference of 2 number is odd
Q = { (1,2), (1,4), (1,6), (2,1), (2,3), (2,5), (3,2), (3,4), (3,6), (4,1), (4,3), (4,5), (5,2), (5,4), (5,6), (6,1), (6,3), (6,5) }

C = P U Q
C = {1,2), (1,4), (1,6), (2,1), (2,2), (2,3), (2,5), (3,2), (3,3), (3,4), (3,6), (4,1), (4,3), (4,5), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,3), (6,5) }

R = The sum of 2 numbers are even
R = {(1,1), (1,3), (1,5), (2,2), (2,4), (2,6), (3,1), (3,3), (3,5), (4,2), (4,4), (4,6), (5,1), (5,3), (5,5), (6,2(, (6,4), (6,6)}

D = P ∩ R
D = {(2,2), (3,3), (3,5), (5,3), (5,5)}

Part 4 (a)
In this case, you need to toss dice and record your result in the table. Just in case you don't have dice at home, you can use these digital dice to do your experiment.

http://www-cs-students.stanford.edu/~nick/settlers/
http://leepoint.net/notes-java/examples ... ldice.html

The following are the sample data. You should do your experiment and collect your own data. Everybody should have different set of data.

From the table,


(i)
(ii)

(iii)
Part 5
(a)

Question:
Based on Table 1, determine the actual mean, the variance and the standard deviation of the sum of all dots on the turned-up faces by using the formulae given.
(b)

Question:
Compare the mean, variance and standard deviation obtained in Part 4 and Part 5. What can you say about the values? Explain in your own words your interpretation and your understanding of the values that you have obtained and relate your answers to the Theoretical and Empirical Probabilities.

Table below shows the comparison of mean, variance and standard deviation of part 4 and part 5
We can see that, the mean, variance and standard deviation that we obtained through experiment in part 4 are different but close to the theoretical value in part 5.

For mean, when the number of trial increased from n=50 to n=100, its value get closer (from 6.58 to 6.91) to the the theoretical value. This is in accordance to the Law of Large Number. We will discuss Law of Large Number in next section.

Nevertheless, the empirical variance and empirical standard deviation that we obtained i part 4 get further from the theoretical value in part 5. This violate the Law of Large Number. This is probably due to

The sample (n=100) is not large enough to see the change of value of mean, variance and standard deviation.
Law of Large Number is not an absolute law. Violation of this law is still possible though the probability is relative low.

In conclusion, the empirical mean, variance and standard deviation can be different from the theoretical value. When the number of trial (number of sample) getting bigger, the empirical value should get closer to the theoretical value. However, violation of this rule is still possible, especially when the number of trial (or sample) is not large enough.

Part 5 (c)

Question:
If n is the number of times two dice are tossed simultaneously, what is the range of mean of the sum of all dots on the turned-up faces as n changes? Make your conjecture and support your conjecture.

The range of the mean
Conjecture: As the number of toss, n, increases, the mean will get closer to 7. 7 is the theoretical mean.

Image below support this conjecture where we can see that, after 500 toss, the empirical mean become very close to the theoretical mean, which is 3.5. (Take note that this is experiment of tossing 1 die, but not 2 dice as what we do in our experiment)