1) Describe two main differences between classical and empirical probabilities. a. Classical probabilities are based on assumptions; Empirical probabilities are based on observations. b. Classical probabilities do not require an action to take place; Empirical probabilities have to have been “performed”.

2) Gather 16 to 30 coins. Shake and empty bag of coins 10 times and tally up how many head and tails are showing.

Number of coins: 20

* Consider the first toss, what is the observed probability of tossing a head? Of tossing a tail? Reduce to the lowest term.
Tossing a Head: 11 / 20
Tossing a Tail: 9 / 20
The fractions are already in the lowest terms.
* Did any of your repetitions have exactly the same number of heads and tails? Yes

* How many times did this happen? Once…10 heads and 10 tails (toss 5)

* Compute the average number of heads from the ten trials (add up the number of heads and divide it by 10).
11 + 8 +11 + 11+ 10 + 12 + 11 + 12 + 13 + 12 = 111
111 / 10 = 11.1
* Change this to the average probability of tossing heads by putting the average number of heads in a fraction over the number of coins you used in your tosses.

11.1 / 20 = 0.555

* Did anything surprising or unexpected happen in your results for this experiment? Yes, I did not expect to so many of the same results: 11H and 9T…4 times
12H and 8T…2 times

3) Write the sample space for the outcomes of tossing three coins using H for heads and T for tails. H: headsT: tails
(HHH, HHT, HTT, HTH, TTT, TTH, THT, THH)
P(E) = n(E)
n(S)
P(E) = ⅛
This is known as a classical probability method.

...PROBABILITY DISTRIBUTION
In the world of statistics, we are introduced to the concept of probability. On page 146 of our text, it defines probability as "a value between zero and one, inclusive, describing the relative possibility (chance or likelihood) an event will occur" (Lind, 2012). When we think about how much this concept pops up within our daily lives, we might be shocked to find the results. Oftentimes, we do not think in these terms, but imagine what the probability of us getting behind the wheel of a car twice a day, Monday through Friday, and arriving at work and home safely. Thankfully, the probability for me has been 'one'! This means that up to this point I have made it to work and returned home every day without getting into an accident. While probability might have one outcome with one set of circumstances, this does not mean it will always turn out that way. Using the same example, just because I have arrived at work every day without getting into an accident, this does not mean it will always be true. As I confess with my words, and pray it does stay the same, probability tells me there is room for a different outcome.
In business, we often look at the probability of success or financial gain when making a decision. There are several things to take into consideration such as the experiment, potential outcomes, and possible events. An...

...of observations, which gives each observation equal weight, the mean of a random variable weights each outcome xi according to its probability, pi. The mean also of a random variable provides the long-run average of the variable, or the expected average outcome over many observations.The common symbol for the mean (also known as the expected value of X) is , formally defined by
Variance - The variance of a discrete random variable X measures the spread, or variability, of the distribution, and is defined by
The standard deviation is the square root of the variance.
Expectation - The expected value (or mean) of X, where X is a discrete random variable, is a weighted average of the possible values that X can take, each value being weighted according to the probability of that event occurring. The expected value of X is usually written as E(X) or m.
E(X) = S x P(X = x)
So the expected value is the sum of: [(each of the possible outcomes) × (the probability of the outcome occurring)].In more concrete terms, the expectation is what you would expect the outcome of an experiment to be on average.
2. Define the following;
a) Binomial Distribution - is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. Therewith the probability of an event is defined by its binomial...

...CHAPTER 3: PROBABILITY DISTRIBUTION
3.1
RANDOM VARIABLES AND PROBABILITY DISTRIBUTION
Random variables is a quantity resulting from an experiment that, by chance, can assume different values. Examples of random variables are the number of defective light bulbs produced during the week and the heights of the students is a class. Two types of random variables are discrete random variables and continuous random variable.
3.2
DISCRETE RANDOM VARIABLE
A random variable is called a discrete random variable if its set of posibble outcomes is countable. Probability distribution is a listing of all the outcomes of an experiment and the probability associated with each outcome. For example, the probability distribution of rolling a die once is as below: Outcome, x Probability, P(x) 1 1 6 2 1 6 3 1 6 4 1 6 5 1 6 6 1 6
The probability distribution for P(x) for a discrete random variable must satisfy two properties: 1. The values for the probabilities must be from 0 to 1; 0 ≤ ( ) ≤ 1 2. The sum for P(x) must be equal to 1; ∑ ( ) = 1
QMT200
3.2.1 FINDING MEAN AND VARIANCE Mean of X is also referred to as its “expected value”.
= ( ) Where: = ∑[ ( )]
( )=
= (
) − [ ( )]
(
)=
[
( )] = ( )
Example 1 An experiment consists of tossing two coins simultaneously. Write down the sample space. If X is the number of tails observed,...

...vs. EmpiricalProbabilityProbability-
describes the chance that an uncertain event will occur.
EmpiricalProbability - estimate that the event will happen based on how often the event occurs after collecting the data or running an experiment. It is based specifically on direct observation or experiences.
EmpiricalProbability Formula
P(E) = probability that an event, E, will occur.
Top = number of ways the specific event occurs.
Bottom = number of ways the experiment could occur.
Example: A survey was conducted to determine students' favorite color. Each student chose only one color.
color Red Blue Black Yellow Purple Other
# 10 15 35 8 5 12
What is the probability that a student's favorite color being black?
Answer: 35 out of the 85 students chose Black. The probability is
.
Theoretical Probability-Theoretical Probability: is the number of ways that an event can occur. You need to divide by the total number of outcomes. This is usually used with equally likely out comes.
Theoretical Probability Formula
P(E) = probability that an event, E, will occur.
n(E) = number of equally likely outcomes of E.
n(S) = number of equally likely outcomes of sample space S. Example 1: Find the probability of rolling...

...Chapter 1
The Problem and Its Background
Introduction
Changes are permanent thing on earth. Are the people is ready enough to accept the changes on the educational system? The current opening of classes here in the Philippines usually starts from June to March but our lawmakers want to amend the opening of classes. The existing school calendar which spans from June to March is often disrupted as destructive typhoons plague the region during the rainy season that’s why our lawmakers decided to move the opening of classes from September through May to avoid numerous class suspensions and serve to protect the students during strong typhoons. The Department of Education said that it is open to the proposal by some sectors, including lawmakers to move the opening of classes but they want to ensure the comfort of the students in school and stresses the need for a comprehensive study.
While the Department of Education is open on the proposal, some did not welcome this idea. An initial survey on the matter conducted way back in 2009. On the respondents, 66 percent were against the move while 34 percent were in favor. [1] Also, another ground for rejecting the proposal is the traditions celebrated during summer and being not conducive in learning because of hot weather during March. The delay in suspension of classes is one of the reasons why our lawmakers and other sectors in the community urge to move the classes from June to September. Unexpected suspension of classes is...

...shareholders.
DIVIDEND SMOOTHING: A concept that has its genesis in the dividend model proposed by John
Lintner (1956).It states that the firms strive towards dividend stability and consistency. The dividend paid
during current year is governed by dividend paid during previous year and variations in the earnings should
not be reflected in the dividend payout.
INFORMATION ASSYMETRY: A situation in which one party in a transaction has more or superior
information compared to another. This often happens in transactions where the seller knows more than the
buyer, although the reverse can happen as well. Potentially, this could be a harmful situation because one
party can take advantage of the other party’s lack of knowledge.
EVENT STUDY: An empirical study performed on a security that has experienced a significant catalyst
occurrence, and has subsequently changed dramatically in value as a result of that catalyst. The event can
have either a positive or negative effect on the value of the security. Event studies can reveal important
information about how a security is likely to react to a given event, and can help predict how other
securities are likely to react to different events.
PECKING ORDER HYPOTHESIS: This hypothesis states that a company which prefers retention of
profits for financing the capital expenditure from internal resources distributes fewer dividends compared
to a firm which finances the investment expenditure from external...

...Probability 2
Theory
Probability theory is the branch of mathematics concerned with probability, the analysis of random phenomena. (Feller, 1966) One object of probability theory is random variables. An individual coin toss would be considered to be a random variable. I predict if the coin is tossed repeatedly many times the sequence of it landing on either heads or tails will be about even.
Experiment
The Experiment we conducted was for ten students to flip a coin one hundred times each and record each time if we got heads or tails.
• The first step was to count the total number of Heads and Tails flipped.
• The second step was to draw circles around consecutive groups of heads and consecutive groups of tails.
• The third step was to count the number of consecutive flips in each cluster.
• The fourth step is to count the clusters.
Analysis
The graph shows the number of consecutive heads or tails from groups of 1 through 10 the coin landed on. On the average, heads was landed on 48.2% of the time, and tails 51.8% of the time.
Conclusion
In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to
Probability...

...Probability
1.) AE-2 List the enduring understandings for a content-area unit to be implemented over a three- to five- week time period. Explain how the enduring understandings serve to contextualize (add context or way of thinking to) the content-area standards.
Unit: Data and Probability
Time: 3 weeks max
Enduring Understanding:
“Student Will Be Able To:
- Know what probability is (chance, fairness, a way to observe our random world, the different representations)
- Know what the difference between experimental and theoretical probability is
- Be able to find the probability of a single event
- Be able to calculate the probability of sequential events, with and without replacement
- Understand what a fair game is and be able to determine if a game is fair
- Be able to make a game fair
- Be able to use different approaches (such as tree diagrams, area models, organized lists) to solve probability problems in life.
- Be able to predict the characteristics of an entire population from a representative sample
- Be able to analyze a representative sample for flaws in its selection
- Be able to create and interpret different statistical representations of data (bar graphs, line graphs, circle graphs, stem-and-leaf)
- Be able to choose an appropriate way to display various sets of data
- Know why the Fundamental Counting Principle works and be...