AP Statistics Curriculum 2007 Prob Basics - Revision history

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Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

Dinov: /* A brief History */

2018-04-13T13:22:37Z

‎A brief History

Dinov: /* A brief History */

2018-04-13T13:17:06Z

‎A brief History

Dinov at 12:28, 13 April 2018

2018-04-13T12:28:20Z

Jenny: /* Elementary Probability */

2010-06-28T19:10:38Z

‎Elementary Probability

Jenny: /* Birthday Paradox */

2010-06-28T19:07:37Z

‎Birthday Paradox

Jenny: /* Axioms of probability */

2010-06-28T19:07:09Z

‎Axioms of probability

Jenny: /* Law of Large Numbers */

2010-06-28T19:06:09Z

‎Law of Large Numbers

Jenny: /* Hands-on activities */

2010-06-28T19:05:01Z

‎Hands-on activities

Jenny: /* Random Sampling */

2010-06-28T19:03:31Z

‎Random Sampling

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	* SOCR Home page: http://www.socr.ucla.edu		* SOCR Home page: http://www.socr.ucla.edu

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	The next substantial contributions to probability theory was a [http://wiki.socr.umich.edu/images/b/bc/Probability_Expectation_Formulation_Pascal_Fermat_1654.pdf letter exchange] by [https://en.wikipedia.org/wiki/Blaise_Pascal Blaise Pascal] (1623-1662) and [https://en.wikipedia.org/wiki/Pierre_de_Fermat Pierre de Fermat] (1601-1665). Pascal and Fermat discussed a gambling problem proposed in 1654 by [https://fr.wikipedia.org/wiki/Antoine_Gombaud,_chevalier_de_M%C3%A9r%C3%A9 Chevalier de Mere], which examined the fundamentals of probability theory. The key question related to the number of experiments required to ensure obtaining a total sum of 6 when rolling two fair hexagonal dice. The letter correspondence between Pascal and Fermat led to the formulation of probability and expectation.		The next substantial contributions to probability theory was a [http://wiki.socr.umich.edu/images/b/bc/Probability_Expectation_Formulation_Pascal_Fermat_1654.pdf letter exchange] by [https://en.wikipedia.org/wiki/Blaise_Pascal Blaise Pascal] (1623-1662) and [https://en.wikipedia.org/wiki/Pierre_de_Fermat Pierre de Fermat] (1601-1665). Pascal and Fermat discussed a gambling problem proposed in 1654 by [https://fr.wikipedia.org/wiki/Antoine_Gombaud,_chevalier_de_M%C3%A9r%C3%A9 Chevalier de Mere], which examined the fundamentals of probability theory. The key question related to the number of experiments required to ensure obtaining a total sum of 6 when rolling two fair hexagonal dice. The letter correspondence between Pascal and Fermat led to the formulation of probability and expectation.
		+
		+	The most popular modern probability theory foundation (as there are many alternative axiomatic definitions) was laid out by [https://en.wikipedia.org/wiki/Andrey_Nikolaevich_Kolmogorov Andrey Kolmogorov], who defined the notion of ''sample space'' and proposed an axiomatic system for probability theory, and [https://en.wikipedia.org/wiki/Richard_von_Mises Richard von Mises], who tied in measure theory.

	=== Fundamentals of Probability Theory===		=== Fundamentals of Probability Theory===

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 Currently, there is no evidence that the ancient Greeks or the Romans had any interests or pursued the notion of predicting future likelihoods, probabilities, outcome chances, or odds. The first recorded attempts to develop a theory describing future events was driven by examining gambling games during hte early Renaissance. In 1550, an Italian polymath [https://en.wikipedia.org/wiki/Gerolamo_Cardano Gerolamo Cardano] (1501-1576) wrote a [https://books.google.com/books/about/Foundations_of_discrete_mathematics.html?id=xHlQAAAAMAAJ paper] addressing the chance of certain outcomes in rolls of dice, which presented the first definition of ''probability''. This manuscript was only in 1576 and then printed in 1663.
-The next substantial contributions to probability theory was a letter exchange by [https://en.wikipedia.org/wiki/Blaise_Pascal Blaise Pascal] (1623-1662) and [https://en.wikipedia.org/wiki/Pierre_de_Fermat Pierre de Fermat] (1601-1665). Pascal and Fermat discussed a gambling problem proposed in 1654 by [https://fr.wikipedia.org/wiki/Antoine_Gombaud,_chevalier_de_M%C3%A9r%C3%A9 Chevalier de Mere], which examined the fundamentals of probability theory. The key question related to the number of experiments required to ensure obtaining a total sum of 6 when rolling two fair hexagonal dice. The letter correspondence between Pascal and Fermat led to the formulation of probability and expectation.
+The next substantial contributions to probability theory was a [http://wiki.socr.umich.edu/images/b/bc/Probability_Expectation_Formulation_Pascal_Fermat_1654.pdf letter exchange] by [https://en.wikipedia.org/wiki/Blaise_Pascal Blaise Pascal] (1623-1662) and [https://en.wikipedia.org/wiki/Pierre_de_Fermat Pierre de Fermat] (1601-1665). Pascal and Fermat discussed a gambling problem proposed in 1654 by [https://fr.wikipedia.org/wiki/Antoine_Gombaud,_chevalier_de_M%C3%A9r%C3%A9 Chevalier de Mere], which examined the fundamentals of probability theory. The key question related to the number of experiments required to ensure obtaining a total sum of 6 when rolling two fair hexagonal dice. The letter correspondence between Pascal and Fermat led to the formulation of probability and expectation.
 === Fundamentals of Probability Theory===

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	==[[AP_Statistics_Curriculum_2007 \| General Advance-Placement (AP) Statistics Curriculum]] - Fundamentals of Probability Theory==		==[[AP_Statistics_Curriculum_2007 \| General Advance-Placement (AP) Statistics Curriculum]] - Fundamentals of Probability Theory==
		+
		+	=== A brief History===
		+	Currently, there is no evidence that the ancient Greeks or the Romans had any interests or pursued the notion of predicting future likelihoods, probabilities, outcome chances, or odds. The first recorded attempts to develop a theory describing future events was driven by examining gambling games during hte early Renaissance. In 1550, an Italian polymath [https://en.wikipedia.org/wiki/Gerolamo_Cardano Gerolamo Cardano] (1501-1576) wrote a [https://books.google.com/books/about/Foundations_of_discrete_mathematics.html?id=xHlQAAAAMAAJ paper] addressing the chance of certain outcomes in rolls of dice, which presented the first definition of ''probability''. This manuscript was only in 1576 and then printed in 1663.
		+
		+	The next substantial contributions to probability theory was a letter exchange by [https://en.wikipedia.org/wiki/Blaise_Pascal Blaise Pascal] (1623-1662) and [https://en.wikipedia.org/wiki/Pierre_de_Fermat Pierre de Fermat] (1601-1665). Pascal and Fermat discussed a gambling problem proposed in 1654 by [https://fr.wikipedia.org/wiki/Antoine_Gombaud,_chevalier_de_M%C3%A9r%C3%A9 Chevalier de Mere], which examined the fundamentals of probability theory. The key question related to the number of experiments required to ensure obtaining a total sum of 6 when rolling two fair hexagonal dice. The letter correspondence between Pascal and Fermat led to the formulation of probability and expectation.

	=== Fundamentals of Probability Theory===		=== Fundamentals of Probability Theory===

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 ===Examples===
 ====Elementary Probability====
-Each of the three boxes below contains two types of balls (''Red'' and ''Green''). Box 1 has ''4 Red'' and ''3 Green'' balls, box 2 has ''3 Red'' and ''2 Green'' balls, and box 3 has ''2 Red'' and ''1 Green'' balls. All balls are identical except for their labels. Which of the three boxes are you most likely to draw a ''Red'' ball from? In other words, if a randomly drawn ball is known to be Red, which box is the one that we most likely drew the ball out of?
+Each of the three boxes below contains two types of balls (''Red'' and ''Green''). Box 1 has ''4 Red'' and ''3 Green'' balls, box 2 has ''3 Red'' and ''2 Green'' balls, and box 3 has ''2 Red'' and ''1 Green'' balls. All balls are identical except for their labels. Which of the three boxes do you most likely to draw a ''Red'' ball from? In other words, if a randomly drawn ball is known to be Red, which box is the one that we most likely draw the ball out of?
 <center>[[Image:SOCR_EBook_Dinov_Prob_BallUrn_040308_Fig4.jpg|400px]]</center>
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 * ''Empirical solution'': This problem may also be explored experimentally using the [http://socr.ucla.edu/htmls/exp/Ball_and_Urn_Experiment.html SOCR Ball-and-Urn Experiment] (see [[SOCR_EduMaterials_Activities_BallAndRunExperiment |this activity]]). Each of the 3 figures below illustrates this experiment, where we sample (''with replacement'') 100 balls from each box, respectively.
-** Box 1: To empirically test the chance of drawing a Red ball from box 1, we set N=7 (total number of balls), R=4 (number of red balls in the population, and n=100 (number of balls we sample, i.e., number of experiments we do). The result of these 100 random draws will vary each time, however one such experiment generated 62 Red balls out of the 100 draws (see image below and the value of the ''Y=62'' variable in the summary table). <center>[[Image:SOCR_EBook_Dinov_Prob_BallUrn_040308_Fig1.jpg|400px]]</center>
+** Box 1: To empirically test the chance of drawing a Red ball from box 1, we set N=7 (total number of balls), R=4 (number of red balls in the population, and n=100 (number of balls we sample, i.e., number of experiments we do). The result of these 100 random draws will vary each time. However one such experiment generated 62 Red balls out of the 100 draws (see image below and the value of the ''Y=62'' variable in the summary table). <center>[[Image:SOCR_EBook_Dinov_Prob_BallUrn_040308_Fig1.jpg|400px]]</center>
-** Box 2: Now we set N=5 (total number of balls), R=3 (number of red balls in the population, and n=100 (number of balls we sample, i.e., number of experiments we do). Again, the result of these 100 random draws will vary each time, however one such experiment generated 64 Red balls out of the 100 draws (see image below and the value of the variable ''Y=64'' in the summary table below the graph). <center>[[Image:SOCR_EBook_Dinov_Prob_BallUrn_040308_Fig2.jpg|400px]]</center>
+** Box 2: Now we set N=5 (total number of balls), R=3 (number of red balls in the population, and n=100 (number of balls we sample, i.e., number of experiments we do). Again, the result of these 100 random draws will vary each time. However one such experiment generated 64 Red balls out of the 100 draws (see image below and the value of the variable ''Y=64'' in the summary table below the graph). <center>[[Image:SOCR_EBook_Dinov_Prob_BallUrn_040308_Fig2.jpg|400px]]</center>
-** Box 3: Finally, we can set N=3 (total number of balls), R=2 (number of red balls in the population, and n=100 (number of balls we sample, i.e., number of experiments we do). Again, the result of these 100 random draws will vary each time, however one such experiment generated 71 Red balls out of the 100 draws (see image below and the value of the variable ''Y=71'' in the summary table below the graph).
+** Box 3: Finally, we can set N=3 (total number of balls), R=2 (number of red balls in the population, and n=100 (number of balls we sample, i.e., number of experiments we do). Again, the result of these 100 random draws will vary each time. However one such experiment generated 71 Red balls out of the 100 draws (see image below and the value of the variable ''Y=71'' in the summary table below the graph).
 <center>[[Image:SOCR_EBook_Dinov_Prob_BallUrn_040308_Fig3.jpg|400px]]</center>

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 Note: for N=42, P>0.9.
-This is an ''approximate solution'' to the Birthday problem, you can also see the ''exact solution'' in the [[SOCR_EduMaterials_Activities_BirthdayExperiment#Exact_Calculations | Birthday Paradox Activity]].
+This is an ''approximate solution'' to the Birthday problem. You can also see the ''exact solution'' in the [[SOCR_EduMaterials_Activities_BirthdayExperiment#Exact_Calculations | Birthday Paradox Activity]].
 ===Examples===

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 * '''First axiom''': The probability of an event is a non-negative real number: <math>P(E)\geq 0</math>, <math>\forall E\subseteq  S</math>, where <math>S</math> is the sample space.
-* '''Second axiom''': This is the assumption of '''unit measure''': that the probability that some elementary event in the entire sample space will occur is 1.  More specifically, there are no elementary events outside the sample space: <math>P(S) = 1</math>. This is often overlooked in some mistaken probability calculations; if you cannot precisely define the whole sample space, then the probability of any subset cannot be defined either.
+* '''Second axiom''': This is the assumption of '''unit measure''': the probability that some elementary event in the entire sample space will occur is 1.  More specifically, there are no elementary events outside the sample space: <math>P(S) = 1</math>. This is often overlooked in some mistaken probability calculations if you cannot precisely define the whole sample space, then the probability of any subset cannot be defined either.
 * '''Third axiom''': This is the assumption of additivity: Any countable sequence of pair-wise disjoint events <math>E_1, E_2, ...</math> satisfies <math>P(E_1 \cup E_2 \cup \cdots) = \sum_i P(E_i).</math>

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 ===Law of Large Numbers===
-When studying the behavior of coin tosses, the law of large numbers implies that the relative proportion (relative frequency) of heads-to-tails in a coin toss experiment becomes more and more stable as the number of tosses increases. This regards the relative frequencies, not absolute counts of heads and tails.
+When studying the behavior of coin tosses, the law of large numbers implies that the relative proportion (relative frequency) of heads-to-tails in a coin toss experiment becomes more and more stable as the number of tosses increases. This regards to the relative frequencies, not absolute counts of heads and tails.
-* There are two widely held ''misconceptions'' about what the law of large numbers about coin tosses:
+* There are two widely held ''misconceptions'' about the law of large numbers relating to coin tosses:
 **Differences between the actual numbers of heads and tails become more variable with increase of the number of tosses – a sequence of 10 heads doesn’t increase the chances of selecting a tail on the next trial.
 ** Coin toss results are independent and fair, and the outcome behavior is unpredictable.

@@ Line 22: / Line 22: @@
 ** '''Swap''' strategy - choose one card first, as the computer reveals one of the donkey cards, you always swap your original guess and go with the third face-down card!
-* You can try the [http://socr.ucla.edu/htmls/SOCR_Experiments.html Monty Hall Experiment], as well. There you can run a very large number of trials automatically and empirically observe the outcomes. Notice that your chance to win doubles if you use the swap-strategy. Why is that?
+* You can try the [http://socr.ucla.edu/htmls/SOCR_Experiments.html Monty Hall Experiment] as well. There you can run a very large number of trials automatically and observe the outcomes empirically. Notice that your chance to win doubles if you use the swap-strategy. Why is that?
 * See the [[SOCR_EduMaterials_Activities_MontyHall | SOCR Monty Hall Activity]].

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 A '''simple random sample''' of ''n'' items is a sample in which every member of the population has an equal chance of being selected ''and'' the members of the sample are chosen independently.
-* '''Example''': Consider a class of students as the population under study.  If we select a sample of size 5, each possible sample of size 5 must have the same chance of being selected. When a sample is chosen randomly, it is the process of selection that is random. How could we select five members from this class randomly? Random sampling from finite (or countable) populations is well-defined. On the contrary, random sampling of uncountable populations is only allowed under the [http://en.wikipedia.org/wiki/Axiom_of_Choice Axiom of Choice].
+* '''Example''': Consider a class of students as the population under study.  If we select a sample of size 5, each possible sample of size 5 must have the same chance of being selected. When a sample is chosen randomly, the process of selection that is random. How could we select five members from this class randomly? Random sampling from finite (or countable) populations is well-defined. On the contrary, random sampling of uncountable populations is only allowed under the [http://en.wikipedia.org/wiki/Axiom_of_Choice Axiom of Choice].
 * '''Random Number Generation''' using [[SOCR]]:  You can use [http://socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler] to [[SOCR_EduMaterials_Activities_RNG | construct random samples]] of any size from a large number of distribution families.
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 * '''Questions''':
 **How would you go about randomly selecting five students from a class of 100?
-**How representative of the population is the sample likely to be? The sample won’t exactly resemble the population as there will be some chance variation.  This discrepancy is called chance '''error due to sampling'''.
+**How likely the sample is to represent of the population? The sample won’t exactly resemble the population as there will be some chance of variation.  This discrepancy is called chance '''error due to sampling'''.
-* '''Definition''': '''Sampling bias''' is non-randomness that refers to some members having a tendency to be selected more readily than others. When the sample is biased the statistics turn out to be poor estimates.
+* '''Definition''': '''Sampling bias''' is non-random and refers to some members having a tendency to be selected more readily than others. When the sample is biased the statistics turn out to be poor estimates.
 ===Hands-on activities===