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 able to use it to solve counting problems.” http://www.arps.org/Curriculum/Maps/MS/Mathematics/Grade7.pdf

2.) AE-3 List the language abilities that ELLs must develop to access the content you are teaching in your unit; then list the language abilities that they need to demonstrate content mastery.

Academic Language Abilities:
* Know the difference between possibility and probability. * Expressing probability and improbability
* Words that have representations of mathematical meaning.

AE- 4 Determine the content-area learning outcomes that all students will master as a result of...

...Learning Programmes Division
Second Semester 2010-2011
Course Handout
Course Number
Course Title
: AAOC ZC111
: Probability and Statistics
Course E-mail address : aaoczc111@dlpd.bits-pilani.ac.in
Course Description
Probability spaces; conditional probability and independence; random variables and probability
distributions; marginal and conditional distributions; independent random variables, mathematical
exceptions, mean and variance, Binomial Poisson and normal distribution; sum of independent random
variables; law of large numbers; central limit theorem; sampling distributions; tests for mean using normal
and student’s distributions; tests of hypotheses; correlation and linear regression.
Scope and Objectives
At the end of the course, the student should be able to understand probabilistic & deterministic models and
statistical inference and apply these concepts to solve a variety of problems.
Prescribed Text Book
T1
Johnson Richard A. & C.B. Gupta, Miller & Freund’s Probability and Statistics for Engineers, PHI,
7th Ed., 2005.
Reference Books
R1.
Paul L. Meyer, Introductory Probability and Statistical Appl., Second Edition. Addison-Wesley, 1970.
R2. M.S. Radhakrishnan, Probability & Statistics, DLPD Notes Note: Softcopy of this Supplementary notes will
be available for download from BITS DLP website.
R3. Mendenhall Beaver Beaver, Introduction to...

...random variable is
A) generated by a random number table.
B) the variable for which an algebraic equation is solved.
C) a numerical measure of a probability experiment.. Ans = C
D) a qualitative attribute of a population.
4) Given the table of probabilities for the random variable x, does this form a probability distribution? Answer yes or no.
x 5 10 15 25
P(x) 0.1 –0.1 0.3 0.8 Ans = No
5) True or False: The expected value of a discrete random variable may be negative Ans = True
6) The table of probabilities of the random variable x is given as:
x 0 1 2 5
P(x) 0.5 0.2 0.2 0.1
Find the mean, µ and standard deviation, σ of x. Round answers to one decimal place. Ans = µ = 1.1, σ = 1.5
7) If p is the probability of success of a binomial experiment then the probability of failure is
A) 1 B) –p C) 1–p D) p + 0.5 Ans = C
8) A binomial experiment has 6 trials with the probability of success on any trial = p = 0.5. Find the probability of exactly 2 successes in the 6 trials. (Use the binomial probability distribution function.) Ans = 0.2344
9) Assume that male and female births are equally likely and the birth of any child does not affect the probability of the gender of any other children. Find the...

...Probability distribution
Definition with example:
The total set of all the probabilities of a random variable to attain all the possible values. Let me give an example. We toss a coin 3 times and try to find what the probability of obtaining head is? Here the event of getting head is known as the random variable. Now what are the possible values of the random variable, i.e. what is the possible number of times that head might occur? It is 0 (head never occurs), 1 (head occurs once out of 2 tosses), and 2 (head occurs both the times the coin is tossed). Hence the random variable is “getting head” and its values are 0, 1, 2. now probability distribution is the probabilities of all these values. The probability of getting 0 heads is 0.25, the probability of getting 1 head is 0.5, and probability of getting 2 heads is 0.25.
There is a very important point over here. In the above example, the random variable had 3 values namely 0, 1, and 2. These are discrete values. It might happen in 1 certain example that 1 random variable assumes 1 continuous range of values between x to y. In that case also we can find the probability distribution of the random variable. Soon we shall see that there are three types of probability distributions. Two of them deal with discrete values of the random variable and one of them deals with continuous...

...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...

...PROBABILITY and MENDELIAN GENETICS LAB
Hypothesis: If we toss the coin(s) for many times, then we will have more chances to reach the prediction that we expect based on the principle of probability.
Results:
As for part 1: probability of the occurrence of a single event, the deviation of heads and tails of 20 tosses is zero, which means that the possibility of heads and tails is ten to ten, which means equally chances. The deviation of heads and tails of 30 tosses is 4, which means that the occurrence of head is 19 and the occurrence of tail is 11. The deviation of heads and tails of 50 tosses is 3, which means that the occurrence of head is 28 and the occurrence of tail is 22. Compare the second and third observation, we can find that the deviation decrease one. It is corresponding to the hypothesis. The more times we use to toss the coin, the more opportunities we will get to reach the prediction based on the principle of probability.
As for part 2: probability of independent events occurring simultaneously, the observation of Heads-Heads is 11, which is 27.5% of the total experiment. And the deviation is 1. The observation of Heads-Tails or Tail-Heads is 16, which is 40% of the total. And the deviation is 4. The observation of Tails- Tails is 13, which is 32.5% of the total number. And the deviation is 3.
Discussion:
In this lab, we’d learn about the likelihood that a particular event will...

...I. Probability Theory
* A branch of mathematics concerned with the analysis of random phenomena. The outcome of a random event cannot be determined before it occurs, but it may be any one of several possible outcomes. The actual outcome is considered to be determined by chance.
* The word probability has several meanings in ordinary conversation. Two of these are particularly important for the development and applications of the mathematical theory ofprobability. One is the interpretation of probabilities as relative frequencies, for which simple games involving coins, cards, dice, and roulette wheels provide examples.
* It is the likeliness of an event happening based on all the possible outcomes. The ratio for the probability of an event 'P' occurring is P (event) = number of favorable outcomes divided by number of possible outcomes.
Example:
A coin is tossed on a standard 8×8 chessboard.
What is the theoretical probability that the coin lands on a black square?
Choices:
A. 0.5
B. 0.25
C. 0.42
D. 0.6
Correct answer: A
Solution:
Step 1: Theoretical probability = number of favorable outcomes / number of possible outcomes.
Step 2: The probability of the coin lands on the black square is 32.
Step 3: Total number of outcomes = 64.
Step 4: P (event) =
Step 5: == 0.5
Step 6: The theoretical probability that...

...Notation for the Binomial Distribution
P(S) The symbol for the probability of success
P(F) The symbol for the probability of failure
p The numerical probability of a success
q The numerical probability of a failure
P(S) = p and P(F) = 1 - p = q
n The number of trials
X The number of successes
The probability of a success in a binomial experiment can be computed with the following formula.
Binomial Probability Formula
In a binomial experiment, the probability of exactly X successes in n trials is
An explanation of why the formula works will be given in the following example.
Example 1:
A coin is tossed three times. Find the probability of getting exactly two heads.
Solution:
This problem can be solved by looking that the sample space. There are three ways to get two heads.
HHH, HHT, HTH, THH, TTH, THT, HTT, TTT
The answer is or 0.375.
The probability of a success in a binomial experiment can be computed with the following formula.
Binomial Probability Formula
In a binomial experiment, the probability of exactly X successes in n trials is
An explanation of why the formula works will be given in the following example.
Example 1:
A coin is tossed three...

...MATH PORTFOLIO
Modelling Probabilities on games of tennis
Introduction:
In this portfolio I shall investigate the different models and probabilities based on the probabilities in the game of tennis. First I will start with the Part 1 of the portfolio where I will be concluding with the expected value and the standard distribution from my results.
I will then take a look at the Non Extended play games where the highest of 7 points can be played. This is will be done with the use of binomial distribution. Then I will calculate the odds of Adam winning the game of tennis and will generalize my model so that I can apply to any other player. After making this model, I will take a look at the extended games where in theory the game could go on forever. This is the stage where I find a model to find the odds of Adam winning the extended games and then will generalize this also.
In the Part 3, I will also test the model for different values of point winning probabilities and find out the odds for each of them and then will then look for patterns from the values of odds that I find.
Finally, I will evaluate the benefits and limitations of the models I come up with.
Part 1:
According to the question, Adam wins twice as many points as Ben does in the game of tennis; therefore, the ratio of points won by Adam to Ben is 2:1 respectively. This shows that Adam wins 23of the points and Ben wins 13of the...