Parañaque City

A Term Paper
Presented to:
Edmar Orata

Probability by: Jirolyn Fabro
Miguel Angelo Rosales

March 15, 2012

I - Introduction
II - Interpretations
III - Etymology
IV - History
V - Applications
1. Weather Forecasting
2. Batting Average
3. Winning the Lottery
4.
VI - Discussion
VII -

I-Introduction
Probability is the ratio of the number of ways an event can occur to the number of possible outcomes. Probability is expressed as a fraction or decimal from 0 to 1. Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we are not certain.[1] The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The certainty we adopt can be described in terms of a numerical measure and this number, between 0 and 1, we call probability. [2] The higher the probability of an event, the more certain we are that the event will occur. Thus, probability in an applied sense is a measure of the likeliness that a (random) event will occur.

The concept has been given an axiomatic mathematical derivation in probability theory, which is used widely in such areas of study as mathematics, statistics, finance, gambling, science, artificial intelligence/machine learning and philosophy to, for example, draw inferences about the likeliness of events. Probability is used to describe the underlying mechanics and regularities of complex systems.

Interpretations
The word probability does not have a singular direct definition for practical application. In fact, there are several broad categories ofprobability interpretations, whose adherents possess different (and sometimes conflicting) views about the fundamental nature of probability. For example:
Frequentists talk about probabilities only when dealing with experiments that are random and well-defined. The probability of a random event denotes the relative frequency of occurrence of an experiment's outcome, when repeating the experiment. Frequentists consider probability to be the relative frequency "in the long run" of outcomes.

Subjectivists assign numbers per subjective probability, i.e., as a degree of belief.

Bayesians include expert knowledge as well as experimental data to produce probabilities. The expert knowledge is represented by a prior probability distribution. The data is incorporated in a likelihood function. The product of the prior and the likelihood, normalized, results in a posterior probability distribution that incorporates all the information known to date.

Etymology
The word Probability derives from the Latin probabilitas, which can also mean probity, a measure of the authority of a witness in a legal casein Europe

History
The scientific study of probability is a modern development. Gambling shows that there has been an interest in quantifying the ideas of probability for millennia, but exact mathematical descriptions arose much later. There are reasons of course, for the slow development of the mathematics of probability. Whereas games of chance provided the impetus for the mathematical study of probability, fundamental issues are still obscured by the superstitions of gamblers.
According to Richard Jeffrey, "Before the middle of the seventeenth century, the term 'probable' (Latin probabilis) meant approvable, and was applied in that sense, univocally, to opinion and to action. A probable action or opinion was one such as sensible people would undertake or hold, in the circumstances.However, in legal contexts especially; 'probable' could also apply to propositions for which there was good evidence.

The theory of errors may be traced back to Roger Cotes's Opera Miscellanea (posthumous, 1722), but a memoir prepared by Thomas Simpson in 1755 (printed 1756) first applied the theory to the discussion of errors of observationThe reprint (1757) of this memoir lays down the axioms that positive and negative errors are equally probable, and that certain assignable limits define the range of all errors. Simpson also discusses continuous errors and describes a probability curve.

The first two laws of error that were proposed both originated with Pierre-Simon Laplace. The first law was published in 1774 and stated that the frequency of an error could be expressed as an exponential function of the numerical magnitude of the error, disregarding sign. The second law of error was proposed in 1778 by Laplace and stated that the frequency of the error is an exponential function of the square of the error The second law of error is called the normal distribution or the Gauss law. "It is difficult historically to attribute that law to Gauss, who in spite of his well-known precocity had probably not made this discovery before he was two years old.

Application
Probability theory is applied in everyday life in risk assessment and in trade on commodity markets. Governments typically apply probabilistic methods in environmental regulation, where it is called pathway analysis. 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 neither assessed independently nor necessarily very rationally. The theory of behavioral 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 profoundly affected 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 automobiles and consumer electronics, use reliability theory in product design to reduce the probability of failure. Failure probability may influence a manufacture's decisions on a product's warranty.

The cache language model and other statistical language models that are used in natural language processing are also examples of applications of probability theory.

1. Weather Forecasting
Suppose you have some outdoor plans made for a particular day and the weather report says that the chance of rain is 70%. Should you still go ahead with your plans or should you cancel them for another day? Where does this forecast comes from? Meteorologist are able to calculate the likelihood of what the weather may be on a particular day by looking back in a historical database and examining all the other days in the past that had the same weather characteristics and then determine that on 70% of those similar days it rained. The mathematical formula for probability can be used to demonstrate these findings. When looking for the chance it will rain, this will be the number of days in the database that it rained is divided by the total number of similar days. For example, if there is data for 100 days with similar weather conditions (the sample space), and on 70 of these days it rained (a favorable outcome), the probability of rain on the next similar day is 70/100 or 70%. Since a 50% probability means that an event is just as likely to happen as not to happen, a 70% chance means that it is more likely to rain than not. Therefore, perhaps it is best that you stay home and reschedule your plans for another day!

2. Batting Average
A batting average involves calculating the probability of a player hitting the ball. The sample space is the total number of time a player has had at bat and each hit is a favorable outcome. Therefore, in 10 at-bats a player gets 3 hits, his or her batting average is 3/10 or 30%. For baseball stats, all the percentages are multiplied by 10, so a 30% probability translates to a 300 batting average. So let's say your favorite baseball player is batting 300. This means that when he or she goes up to the plate, they only have a 30% chance of hitting the ball!

3. Winning the Lottery
Millions of people around the world spend their money on lottery tickets in hopes of winning the big jackpot and become millionaires. But do these people realize how low their chances of winning actually are? Determining the probability of winning the lottery will allow you to see what the likelihood of winning truly is. The following formula is used to figure out the probability of winning Lotto 6/49:

Application
Probability theory is applied in everyday life in risk assessment and in trade on commodity markets. Governments typically apply probabilistic methods in environmental regulation, where it is called pathway analysis. 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 neither assessed independently nor necessarily very rationally. The theory of behavioral 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 profoundly affected 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 automobiles and consumer electronics, use reliability theory in product design to reduce the probability of failure. Failure probability may influence a manufacture's decisions on a product's warranty.[15]
The cache language model and other statistical language models that are used in natural language processing are also examples of applications of probability theory.

II - Discussion

Example 1:
Tossing a coin:
There are two possible outcomes. It can be a head or tail, which are both equally likely.

Example 2:
Tossing dice:
Tossing dice is more complicated than tossing coins, as there are more than two values. If you a throw a single dice, then it can fall six ways, each of which is equally likely if the dice is true. So the probability of getting one particular value is 1/6. If you want two either two values it is 2/6 or 1/3, and so on.
Probability Tree Diagram:
Shows all the possible events. The first event is represented by a dot. From the dot, branches are drawn to represent all possible outcomes of the event.
A coin and a dice are thrown at random. Find the probability of
a) getting a head and an even number
b) getting a head or tail and an odd number

A) Getting a head and an even number
Let A denote the event of a head and an even number.
A= ((H, 2), (H, 4), (H, 6)} and n(A) = 3
The chance of getting a Head and an even number is 3/12 or 1/4
B) Getting a head or tail and an odd number
Let B denote the event a head or tail and an odd number.
B= {(H, 1), (H, 3), (H, 5), (T, 1), (T, 3), (T, 5)}

The Chance of getting a Head and an even

Clare tossed a coin three times.
a) Draw a tree diagram to show all the possible outcomes.
b) Find the probability of getting:(i) Three tails.(ii) Exactly two heads.(iii) At least two heads.
Draw a tree diagram to show all the possible outcomes.

B) Find the probability of getting:(i) Three tails.

Let A denote the event of getting three tails.
A={(T,T,T}
The Chance of getting three tails is 1/8.
B) Find the probability of getting:(ii) Exactly two heads.
Let B denote the event of getting exactly two heads.
B={(H,H,T) (H,T,H) (T,H,H)}
The Chance of getting exactly two heads is 3/8.
B) Find the probability of getting:(iii) Exactly two tails.

Let C denote the event of getting exactly two tails.
B={(H,T,T) (T,H,T) (T,T,H)}
The Chance of getting exactly two tails is 3/8.

jiro kung may mga explanation na yung mga example natin kung paano nakuha okay na.. si ko kasi ma identify kung meron na.
Then
wala pa tayong insight at personal evaluation di pa yan naka-ayos
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Dagdagan mo nalang kung gusto mo para dumami maghahanap pa ako ng dadag..
Separate ang paper bawat chapter
Introduction
Discussion
Insight
Evaluation
At first, I was a bit hesitant when I learned that he will be my professor in my favorite subject which is Mathematics. But as the days go by, I’m beginning to appreciate his way of teaching in which I might consider to adapt when I finished my course and be a professor someday. Here’s what I observed from him:
1. He always interacts with the whole class and sees to it that every student understands the topic he discussed.

2. Timely response from him whenever me and my classmates need something to clarify. Approachable indeed.

3. His being specific to the subject matter helped us to be more focus and tends to easily grasp every topic he discussed in our class.

4. He is open to suggestions specially when our class is dealing with a very complex topic. Always finding ways to help us understand the subject matter which is of course a very rare attribute coming from a professor.

5. Since we appreciate his effort in helping us understand his subject, we reciprocate by giving him good comments and in return he doubled his effort in helping us provide better techniques in solving mathematical problems.

6. He always wanted to be criticized so that he can improve his way of teaching and also his personality to the next level. This is seldom found in a professor

7. Being nice to his class is one of the few traits I admire about him. Aside from that, proper grooming and good personal hygiene are among his good traits.

8. His mastery of the subject is what makes him of the best professors I’ve ever had. I’m being biased since my favorite subject is Match, but aside from the way he handles his students and the ways/techniques he taught us made him qualified as such.

9. I could be happier if the professors in all of my subjects are like him or at least half of it. Thank you my professor and congrats for a job well done.