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PROJECT WORK FORADDITIONAL MATHEMATHICS
-2010-
Probability and theirApplication in Our Daily Life
Name Ahmad Firdaus b. Shariffudin
Class 5 Jaya
I/C
Teacher Miss Suriani
School Sekolah Menengah SainsAlam Shah
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CONTENT
Objective
Part 1
Part 2
Part 3Part4
Part5
Further Exploration
Reflection
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Objective
The aims carrying out this project work are:
i. To apply and adapt a variety of problem-solving strategies to solve
problems;
ii. To improve thinking skills;
iii. To promote effective mathematical communication;
iv. To develop mathematical knowledge through problem solving in a
way that increases students interest and confidence;
v. To use the language of mathematics to express mathematical ideas
precisely;
vi. To provide learning environment that stimulates and enhances
effective learning;
vii.To develop positive attitude towards mathematics.
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INTRODUCTION
What is Probability
Probability is a way of expressing knowledge or belief that an event will occur or
has occurred. In mathematicsthe concept has been given an exact meaning
in probability theory, that is used extensively in such areas of study as
mathematics, statistics, finance,gambling, science, and philosophy to draw
conclusions about the likelihood of potential events and the underlying
mechanics ofcomplex systems.
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PART 1
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Theory of Probability
History of Probability
Probability has a dual aspect: on the one hand the probability or likelihood of
hypotheses given the evidence for them, and on the other hand the behavior
ofstochastic processes such as the throwing of dice or coins. The study of the former is
historically older in, for example, the law of evidence, while the mathematical treatment
of dice began with the work ofPascaland Fermat in the 1650s.
Probability is distinguished from statistics. While statistics deals with data and
inferences from it, (stochastic) probability deals with the stochastic (random) processes
which lie behind data or outcomes.
Some highlight in the history of probability are:
18th century: Jacob Bernoulli'sArs Conjectandi(posthumous, 1713) and Abraham de
Moivre's The Doctrine of Chances (1718) put probability on a sound mathematical
footing, showing how to calculate a wide range of complex probabilities. Bernoulli
proved a version of the fundamental law of large numbers, which states that in a large
number of trials, the average of the outcomes is likely to be very close to the expected
value - for example, in 1000 throws of a fair coin, it is likely that there are close to 500
heads (and the larger the number of throws, the closer to half-and-half the proportion is
likely to be).
19th century: The power of probabilistic methods in dealing with uncertainty was shown
by Gauss's determination of the orbit ofCeres from a few observations. The theory of
errors used the method of least squaresto correct error-prone observations, especially
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in astronomy, based on the assumption of a normal distribution of errors to determine
the most likely true value.
Towards the end of the nineteenth century, a major success of explanation in terms of
probabilities was theStatistical mechanics ofLudwig Boltzmann and J. Willard
Gibbs which explained properties of gases such as temperature in terms of the random
motions of large numbers of particles.
The field of the history of probability itself was established by Isaac Todhunter's
monumental History of the Mathematical Theory of Probability from the Time of Pascal
to that of Lagrange (1865).
20th century: Probability and statistics became closely connected through the work
on hypothesis testing ofR. A. FisherandJerzy Neyman, which is now widely applied in
biological and psychological experiments and in clinical trials of drugs. A hypothesis, for
example that a drug is usually effective, gives rise to a probability distribution that would
be observed if the hypothesis is true. If observations approximately agree with the
hypothesis, it is confirmed, if not, the hypothesis is rejected. [5]
The theory of stochastic processes broadened into such areas as Markovprocesses and Brownian motion, the random movement of tiny particles suspended in afluid. That provided a model for the study of random fluctuations in stock markets,
Application of Probability in Daily life
Two major applications of probability theory in everyday life are in risk assessment and
in trade on commodity markets. Governments typically apply probabilistic methods
in environmental regulation where it is called "pathway analysis", often measuring well-
being using methods that are stochastic in nature, and choosing projects to undertake
based on statistical analyses of their probable effect on the population as a whole.
A good example is the effect of the perceived probability of any widespread Middle East
conflict on oil prices - which have ripple effects in the economy as a whole. An
assessment by a commodity trader that a war is more likely vs. less likely sends prices
up or down, and signals other traders of that opinion. Accordingly, the probabilities are
not assessed independently nor necessarily very rationally. The theory ofbehavioral
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http://en.wikipedia.org/wiki/Normal_distributionhttp://en.wikipedia.org/wiki/Statistical_mechanicshttp://en.wikipedia.org/wiki/Ludwig_Boltzmannhttp://en.wikipedia.org/wiki/J._Willard_Gibbshttp://en.wikipedia.org/wiki/J._Willard_Gibbshttp://en.wikipedia.org/wiki/Isaac_Todhunterhttp://en.wikipedia.org/wiki/Isaac_Todhunterhttp://en.wikipedia.org/wiki/Statistical_hypothesis_testinghttp://en.wikipedia.org/wiki/Ronald_Fisherhttp://en.wikipedia.org/wiki/Jerzy_Neymanhttp://en.wikipedia.org/wiki/Clinical_trialshttp://en.wikipedia.org/wiki/Probability_distributionhttp://en.wikipedia.org/wiki/History_of_probability#cite_note-4http://en.wikipedia.org/wiki/Markov_processhttp://en.wikipedia.org/wiki/Markov_processhttp://en.wikipedia.org/wiki/Brownian_motionhttp://en.wikipedia.org/wiki/Riskhttp://en.wikipedia.org/wiki/Commodity_marketshttp://en.wikipedia.org/wiki/Environmental_regulationhttp://en.wikipedia.org/w/index.php?title=Pathway_analysis&action=edit&redlink=1http://en.wikipedia.org/wiki/Measuring_well-beinghttp://en.wikipedia.org/wiki/Measuring_well-beinghttp://en.wikipedia.org/wiki/Stochastichttp://en.wikipedia.org/wiki/Behavioral_financehttp://en.wikipedia.org/wiki/Normal_distributionhttp://en.wikipedia.org/wiki/Statistical_mechanicshttp://en.wikipedia.org/wiki/Ludwig_Boltzmannhttp://en.wikipedia.org/wiki/J._Willard_Gibbshttp://en.wikipedia.org/wiki/J._Willard_Gibbshttp://en.wikipedia.org/wiki/Isaac_Todhunterhttp://en.wikipedia.org/wiki/Statistical_hypothesis_testinghttp://en.wikipedia.org/wiki/Ronald_Fisherhttp://en.wikipedia.org/wiki/Jerzy_Neymanhttp://en.wikipedia.org/wiki/Clinical_trialshttp://en.wikipedia.org/wiki/Probability_distributionhttp://en.wikipedia.org/wiki/History_of_probability#cite_note-4http://en.wikipedia.org/wiki/Markov_processhttp://en.wikipedia.org/wiki/Markov_processhttp://en.wikipedia.org/wiki/Brownian_motionhttp://en.wikipedia.org/wiki/Riskhttp://en.wikipedia.org/wiki/Commodity_marketshttp://en.wikipedia.org/wiki/Environmental_regulationhttp://en.wikipedia.org/w/index.php?title=Pathway_analysis&action=edit&redlink=1http://en.wikipedia.org/wiki/Measuring_well-beinghttp://en.wikipedia.org/wiki/Measuring_well-beinghttp://en.wikipedia.org/wiki/Stochastichttp://en.wikipedia.org/wiki/Behavioral_finance -
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finance emerged to describe the effect of such groupthink on pricing, on policy, and on
peace and conflict.
It can reasonably be said that the discovery of rigorous methods to assess and combine
probability assessments has had a profound effect on modern society. Accordingly, it
may be of some importance to most citizens to understand how odds and probability
assessments are made, and how they contribute to reputations and to decisions,
especially in a democracy.
Another significant application of probability theory in everyday life is reliability. Many
consumer products, such as automobilesand consumer electronics, utilize reliability
theory in the design of the product in order to reduce the probability of failure. The
probability of failure may be closely associated with the product's warranty.
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Theorical Probabilities and Empirical Probabilities
Theorical Probabilities:
Probability theory is the branch ofmathematics concerned with analysis
ofrandom phenomena.[1] The central objects of probability theory are random
variables, stochastic processes, and events: mathematical abstractions ofnon-
deterministic events or measured quantities that may either be single occurrences or
evolve over time in an apparently random fashion. Although an individual coin toss or
the roll of a die is a random event, if repeated many times the sequence of random
events will exhibit certain statistical patterns, which can be studied and predicted. Two
representative mathematical results describing such patterns are the law of large
numbers and the central limit theorem.
As a mathematical foundation forstatistics, probability theory is essential to many
human activities that involve quantitative analysis of large sets of data. Methods of
probability theory also apply to descriptions of complex systems given only partial
knowledge of their state, as instatistical mechanics. A great discovery of twentieth
century physics was the probabilistic nature of physical phenomena at atomic scales,
described in quantum mechanics.
Empirical Probabilities
Empirical probability, also known as relative frequency, orexperimental
probability, is the ratio of the number favorable outcomes to the total number of trials, [1]
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[2] not in a sample space but in an actual sequence of experiments. In a more general
sense, empirical probability estimates probabilities from experience and observation.[3] The phrase a posteriori probability has also been used as an alternative to
empirical probability or relative frequency.[4] This unusual usage of the phrase is not
directly related to Bayesian inference and not to be confused with its equally occasional
use to refer to posterior probability, which is something else.
In statistical terms, the empirical probability is an estimate of a probability. If modelling
using a binomial distributionis appropriate, it is themaximum likelihood estimate. It is
the Bayesian estimate for the same case if certain assumptions are made for the prior
distribution of the probability
An advantage of estimating probabilities using empirical probabilities is that this
procedure is relatively free of assumptions. For example, consider estimating the
probability among a population of men that they satisfy two conditions: (i) that they are
over 6 feet in height; (ii) that they prefer strawberry jam to raspberry jam. A direct
estimate could be found by counting the number of men who satisfy both conditions to
give the empirical probability the combined condition. An alternative estimate could be
found by multiplying the proportion of men who are over 6 feet in height with the
proportion of men who prefer strawberry jam to raspberry jam, but this estimate relies
on the assumption that the two conditions are statistically independent.
A disadvantage in using empirical probabilities arises in estimating probabilities which
are either very close to zero, or very close to one. In these cases very large sample
sizes would be needed in order to estimate such probabilities to a good standard of
relative accuracy. Herestatistical modelscan help, depending on the context, and in
general one can hope that such models would provide improvements in accuracy
compared to empirical probabilities, provided that the assumptions involved actually do
hold. For example, consider estimating the probability that the lowest of the daily-
maximum temperatures at a site in February in any one year is less zero degrees
Celsius. A record of such temperatures in past years could be used to estimate this
probability. A model-based alternative would be to select of family ofprobabilitydistributions and fit it to the dataset contain past yearly values: the fitted distribution
would provide an alternative estimate of the required probability. This alternative
method can provide an estimate of the probability even if all values in the record are
greater than zero.
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Difference between Empirical and Theoretical Probabilities
Empirical probability is the probability a person calculates from many different trials. For
example someone can flip a coin 100 times and then record how many times it came up heads
and how many times it came up tails. The number of recorded heads divided by 100 is the
empirical probability that one gets heads.
The theoretical probability is the result that one should get if an infinite number of trials were
done. One would expect the probability of heads to be 0.5 and the probability of tails to be 0.5
for a fair coin.
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PART 2
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Part 2
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Question:a)Suppose you are playing monopoly game with two of your friends. Tostart the game, each player will have to toss the dice once. The player who
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obtain number will start the game. List all the possible outcomes when thedice is tossed once.
SolutionThere are three player, considered as P1,P2, and P3. The total side of thedie which is cube is six, and the number of dots on the dice is 1, 2, 3, 4, 5and 6 respectively.Thus, the possible outcomes are:{1,2,3,4,5,6}
Question:b) Instead of one die, two dice can also be tossed simultaneously by each
player. The player will move the token according to the sum of all dots onboth turned-up faces. For example, if two dice are tossed simultaneouslyand 2 appears on one dice and 3 on the other, the outcome of the tossis (2,3). Hence, the player shall move the token 5 spaces. Notes: Theevents (2,3) and (3,2) should be treated as two different events.
List all the possible outcomes when two dice are tossed simultaneously.Organize and present your list clearly. Consider the use of table, chart oreven diagram.
Solution
By tossing two dice, the total possible outcomes are:{(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)}
OR
By using table, the possible outcomes when two dice are tossed can belisted.
1 2 3 4 5 6
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1 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
2 (2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
3 (3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
4 (4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
5 (5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
6 (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
The total possible outcomes from the tossing of the two dice is 36, or
n(S)=6X6=36, which are applied from the multiplication rule.
OR
OR
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PART 3
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Part 3
Question:The Table 1 shows the sum of all dots on both turned up faces when twodice are tossed simultaneously.(a) Complete Table 1 by listing all possible outcomes and their
corresponding probabilities.
Sum of the dots on
both turned up faces(x)
Possible outcomes Probability,p(x)
1 - 0
2 (1,1) 1/36
3 (1,2), (2,1) 1/18
4 (1,3), (2,2), (3,1) 1/12
5 (1,4), (2,3), (3,2), (4,1) 1/9
6 (1,5), (2,4), (3,3), (4,2), 5,1) 5/36
7 (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) 1/6
8 (2,6), (3,5), (4,4), (5,3), (6,2) 5/36
9 (3,6), (4,5), (5,4), (6,3) 1/9
10 (4,6), (5,5), (6,4) 1/12
11 (5,6), (6,5) 1/18
12 (6,6) 1/36
Total 36 1
(b) Based on Table 1 that you have competed, list all the possibleoutcomes of the following events and hence find their correspondingprobabilities:A= {The two numbers are not the same}
B= {The product of the two numbers is greater than 36}C= {Both numbers are prime or the difference between twonumbers
is odd}D={The sum of the two numbers are even and both numbers are
prime}
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Solution
1 2 3 4 5 6
1 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
2 (2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
3 (3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
4 (4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
5 (5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
6 (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
A={ (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2),(3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4),(5,6), (6,1), (6,2), (6,3), (6,4), (6,5)}P(A)=??A={(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}P(A)=1/6As P(A)=P(A)=1/6, thus P(A)=1-1/6
=5/6B={},as the maximum product is 6X6=36. This event is impossible to occur.Thus,P(B)=0Prime number(below six):2,3,5Odd number(below six):1,3,5
C = P U QC={(1,2), (1,4), (1,6), (2,1), (2,2), (2,3), (2,5), (3,2), (3,3), (3,4), (3,5), (3,6),(4,1), (4,3), (4,5), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,3), (6,5)}
=23/36
D = P RD={ (2,2), (3,3), (3,5), (5,3), (5,5)}
P(D) =5/36
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Answers:A={ (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2),(3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4),(5,6), (6,1), (6,2), (6,3), (6,4), (6,5)}P(A)= 5/6
B={}P(B)=0
C={(1,2), (1,4), (1,6), (2,1), (2,2), (2,3), (2,5), (3,2), (3,3), (3,4), (3,5), (3,6),(4,1), (4,3), (4,5), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,3), (6,5)}P(C)= 23/36
D={ (2,2), (3,3), (3,5), (5,3), (5,5)}
P(D) =5/36
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PART 4
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Part 4
(a) Conduct an activity by tossing two dice simultaneously 50times. Observe the sum of all dots on both turned up faces.Complete the frequency table below.
Sum of the twonumbers(x)
Frequency( )
2 1 2 4
3 2 6 18
4 5 20 805 3 15 75
6 6 36 216
7 10 70 490
8 8 64 512
9 6 54 486
10 6 60 600
11 2 22 242
12 1 12 144
Total 50 361 2867
Based on Table 2 that you have completed,determine the value of:
MeanVariance: andStandard deviation
Of the data
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Solution,
From the table,
i)
mean, =
ii)
variance,
=
=5.2116
iii)
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Standard deviation,
=
= 2.2829
b) Predict the value if the mean if the number of tosses isincreased to 100 times.
-the number of tosses is increased double, the mean will slightly
change, maybe will inducted by 2.
New mean,
c) Test your prediction in (b) by continuing Activity 3(a) until thetotal number of tosses is 100 times. Then, determine the valueof:
i)mean
ii)variance: and
iii)standard deviation
of the new data.
Was your prediction proved?
Solution:
Sum of the twonumbers(x)
Frequency( )
2 5 10 20
3 5 15 45
4 10 40 160
5 9 45 225
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6 15 90 540
7 16 112 784
8 14 112 896
9 13 117 1053
10 6 60600
11 5 55 605
12 2 24 288
Total 100 680 5216
Solution,
From the table,
mean, =
variance,
=
=5.92
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Standard deviation,
=
= 2.4331
The prediction is wrong. The new mean is 6.8, which 0.42
lesser than the original mean.
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PART 5
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Part 5
When two dice are tossed simultaneously, the actual meanand variance of the sum of all dots on the turned-up
faces can be determined by using the formulae below:
Mean=
Variance=
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 formula given.
Compare the mean, variance and standard deviation
obtained in Part 4 and Part 5. What can you say about
the values? Explain in your words your interpretation
and your understanding of the values that you haveobtained and relate your answers to the Theorical and
Empirical Probabilities
If n is the number of times of two dice are tossed
simultaneously, what is the range of mean of all dots on
the turned-up faces as n changes? Make your conjecture
and support your conjucture.
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Solution:
Sum of the dots onboth turned up faces(x)
Possible outcomes Probability,p(x)
1 - 02 (1,1) 1/36
3 (1,2), (2,1) 1/18
4 (1,3), (2,2), (3,1) 1/12
5 (1,4), (2,3), (3,2), (4,1) 1/9
6 (1,5), (2,4), (3,3), (4,2), 5,1) 5/36
7 (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) 1/6
8 (2,6), (3,5), (4,4), (5,3), (6,2) 5/36
9 (3,6), (4,5), (5,4), (6,3) 1/9
10 (4,6), (5,5), (6,4) 1/12
11 (5,6), (6,5) 1/18
12 (6,6) 1/36
Total 36 1
(a) i) Mean=
+12
=7
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ii) Variance=
+144 ]-
=54.8333-49
=5.8333
iii) standard deviation,
=
=2.4152
The mean, variance and the standard deviation of data in
Part 4 and Part 5 are totally different. Mean, variance,and standard deviation of the data in Part 5 exceeds the
mean, variance, and standard deviation of the data in
Part 4 by o.44, 0.0857, and 0.0179 respectively. The
values are different because there are two different
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method used to identify the mean, variance, and
standard deviation which are by conducting an
experiment as conducted in Part 4 and by using
formulae in Part 5. In Part 4, the values may varies asthe result from the tossing of the dice are always
different. The probability to always get the same number
are very small, which is 1/36. Thus, it affect the values of
the mean, variance, and standard deviation of the data.
The method used in Part 4 to obtain these values also
known as Empirical Probabilities experiment.
Theoretical probabilities are used in identifying thosedata in Part 5. The data are obtained from the
formula and the data will be constant as it is only
theoretical.
Conjecture: As the number of n increases, the meanwill become closer to the theoretical mean, which are
7.00.
Support and proof
From the part 4 experiment, it is obvious that when the
number of n increases, which are 100, the mean become
closer to 7 than when the value of n 50.
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FURTHER
EXPLORATION
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Further Exploration
In probability theory, the Law of Large Numbers
(LNN) is a theorem that describes the result of
performing the same experiment a large number of
times. Conduct a research using the internet to find out
the theory of LLN. When you have finished with your
research, discuss and write about your findings. Relate
the experiment that you have done in this project to theLLN.
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Answer:
In probability theory, the law of large numbers (LLN) is atheorem that describes the
result of performing the same experiment a large number of times. According to the law,
theaverage of the results obtained from a large number of trials should be close to
the expected value, and will tend to become closer as more trials are performed.
For example, a single roll of a six-sided die produces one of the numbers 1, 2, 3, 4, 5, 6,
each with equal probability. Therefore, the expected value of a single die roll is
According to the law of large numbers, if a large number of dice are rolled, the
average of their values (sometimes called the sample mean) is likely to be close to
3.5, with the accuracy increasing as more dice are rolled.
Similarly, when a fair coin is flipped once, the expected value of the number of
heads is equal to one half. Therefore, according to the law of large numbers, the
proportion of heads in a large number of coin flips should be roughly one half. In
particular, the proportion of heads aftern flips will almost surelyconverge to one
half as n approaches infinity.
Though the proportion of heads (and tails) approaches half, almost surely the
absolute (nominal) difference in the number of heads and tails will become large as
the number of flips becomes large. That is, the probability that the absolute
difference is a small number approaches zero as number of flips becomes large.
Also, almost surely the ratio of the absolute difference to number of flips will
approach zero. Intuitively, expected absolute difference grows, but at a slower rate
than the number of flips, as the number of flips grows.
The LLN is important because it "guarantees" stable long-term results for random
events. For example, while a casino may lose money in a single spin of
the roulette wheel, its earnings will tend towards a predictable percentage over a
large number of spins. Any winning streak by a player will eventually be overcome
by the parameters of the game. It is important to remember that the LLN only
applies (as the name indicates) when a large numberof observations are
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http://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Theoremhttp://en.wikipedia.org/wiki/Averagehttp://en.wikipedia.org/wiki/Expected_valuehttp://en.wikipedia.org/wiki/Dicehttp://en.wikipedia.org/wiki/Probabilityhttp://en.wikipedia.org/wiki/Sample_meanhttp://en.wikipedia.org/wiki/Fair_coinhttp://en.wikipedia.org/wiki/Almost_surelyhttp://en.wikipedia.org/wiki/Limit_of_a_sequencehttp://en.wikipedia.org/wiki/Almost_surelyhttp://en.wikipedia.org/wiki/Roulettehttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Theoremhttp://en.wikipedia.org/wiki/Averagehttp://en.wikipedia.org/wiki/Expected_valuehttp://en.wikipedia.org/wiki/Dicehttp://en.wikipedia.org/wiki/Probabilityhttp://en.wikipedia.org/wiki/Sample_meanhttp://en.wikipedia.org/wiki/Fair_coinhttp://en.wikipedia.org/wiki/Almost_surelyhttp://en.wikipedia.org/wiki/Limit_of_a_sequencehttp://en.wikipedia.org/wiki/Almost_surelyhttp://en.wikipedia.org/wiki/Roulette -
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considered. There is no principle that a small number of observations will converge
to the expected value or that a streak of one value will immediately be "balanced"
by the others.
An illustration of the Law of Large Numbers using die rolls. As the number of
die rolls increases, the average of the values of all the rolls approaches 3.5.
Same goes to the project, as the tosses increases to 100 times, themean become nearer to 7, which the actual value of mean. If the experimentis continue until 200 times of tossing, the mean will become closer to 7.
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REFLECTION
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Reflection
While I conducting the project, I had learned some moral
values that I practice. This project had taught me to
responsible on the works that are given to me to becompleted. This project also had make me felt more
confidence to do works and not to give easily when we
could not find the solution for the question. I also learned to
be more discipline on time, which I was given about a
month to complete these project and pass up to my teacher
just in time. I also enjoy doing this project during my school
holiday as I spend my time with friends to complete thisproject and it had tighten our friendship.