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PROJECT WORK FOR ADDITIONAL MATHEMATHICS -2010-

Probability and their Application in Our Daily Life

Name

Ahmad Firdaus b. Shariffudin

Class

5 Jaya

I/C Teacher

Miss Suriani

School Alam Shah

Sekolah Menengah Sains

1

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CONTENT

Objective Part 1 Part 2 Part 3 Part4 Part5 Further Exploration















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.















INTRODUCTION What is Probability

Probability is a way of expressing knowledge or belief that an event will occur or has occurred. In mathematics mathematicsthe the concept has been given an exact meaning in probability theory, theory , that is used extensively in such areas of study as mathematics, statistics statistics,, finance finance,, gambling gambling,, science science,, and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex of complex systems. systems .















PART 1















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 of stochastic of stochastic 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 of Pascal and Fermat in the 1650s. Probability is distinguished from statistics 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's Bernoulli 's Ars Conjectandi (posthumous, 1713) and Abraham de Moivre's 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, 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

















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 the Statistical mechanics of Ludwig of Ludwig 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 of R. of R. A. Fisher and Fisher andJerzy Jerzy Neyman, 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 Markov processes and Brownian motion, motion , the random movement of tiny particles suspended in a fluid. That provided a model for the study of random fluctuations in stock markets,

Application of Probability in Daily life

















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 democracy.. Another significant application of probability theory in everyday life is reliability reliability.. Many consumer products, such as automobiles and 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 warranty..

















Theorical Probabilities and Empirical Probabilities Theorical Probabilities: Probability theory is the branch of mathematics of mathematics concerned with analysis of random of random phenomena. [1] The central objects of probability theory are random variables,, stochastic processes, variables processes , and events events:: mathematical abstractions of nondeterministic 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. theorem . As a mathematical foundation for statistics statistics,, probability theory is essential to many human activities that involve quantitative analysis of large sets of data. Methods of

















[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 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, probability , which is something else. In statistical terms, the empirical probability is an estimate of a probability. If modelling using a binomial distribution is appropriate, it is themaximum the maximum 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. 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 Here statistical models can help, depending on the context, and in general one can hope that such models would provide improvements in accuracy

















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



































Part 2

































obtain number will start the game. List all the possible outcomes when the dice is tossed once.

Solution There are three player, considered as P1,P2, and P3. The total side of the die which is cube is six, and the number of dots on the dice is 1, 2, 3, 4, 5 and 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 on both turned-up faces. For example, if two dice are tossed simultaneously and “2” appears on one dice and “3” on the other, the outcome of the toss is (2,3). Hence, the player shall move the token 5 spaces. Notes: The events (2,3) and (3,2) should be treated as two different events.

















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



















































Part 3

Question: The Table 1 shows the sum of all dots on both turned up faces when two dice are tossed simultaneously. (a) (a) Compl Complet ete e Table Table 1 by lis listi ting ng all all pos possi sible ble outc outcome omes s and and thei their r corresponding probabilities.

















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)

















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)}



































Part 4

















Solution,

















Standard deviation,

















6 7 8 9

15 16 14 13

90 112 112 117

540 784 896 1053























































































































































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