SOCR Events May2008 C3 S1 - Revision history

IvoDinov: /* Solve counting problems using the Fundamental Counting Principle */

2008-02-08T19:08:42Z

‎Solve counting problems using the Fundamental Counting Principle

IvoDinov: /* Solve counting problems using the Fundamental Counting Principle */

2008-02-08T19:07:41Z

‎Solve counting problems using the Fundamental Counting Principle

IvoDinov at 18:56, 8 February 2008

2008-02-08T18:56:28Z

IvoDinov: New page: == SOCR May 2008 Event - Calculate probabilities of events and compare theoretical and experimental probability== ==Solve counting problems using the Fundamental...

2008-02-08T18:41:22Z

New page: == SOCR May 2008 Event - Calculate probabilities of events and compare theoretical and experimental probability== ==Solve counting problems using the Fundamental...

New page

== [[SOCR_Events_May2008 | SOCR May 2008 Event]] - Calculate probabilities of events and compare theoretical and experimental probability==

==Solve counting problems using the Fundamental Counting Principle==

==Calculate the probability of an event or sequence of events with and without replacement using models ==
The [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html Ball and Urn Experiment] demonstrates the effects of sampling with or without replacement. Suppose we '''choose n balls''' at random from an urn containing '''R are red''' and ('''N-R) are green''' balls. For every trial, the '''number of red balls drawn (Y)''' are recorded numerically in the distribution table (on the right) and graphically in the distribution graph (blue). At each trial, the empirical frequency of Y is displayed in '''red''' the distribution graph. The [[SOCR_EduMaterials_Activities_BallAndRunExperiment | experimenter has the abilities to manipulate trials by choosing with replacement or without replacement in the list box and varying parameters N, R, and n with scroll bars]].
[[Image:SOCR_Activities_BallAndUrnExperiment_Dinov_020808_Fig1.jpg|500px]]

* Why is there a difference in sampling balls from the urn with or without replacement?
* Set N=50, R=25 and n=2. If you sample with replacement, what is the chance that we get 2 red balls in the sample of 2?
* If we sample without replacement and the first ball is red, what is the chance that the second ball will be also red? Is this probability effected by knowing the collor of the first drawn ball?
* If we increase or decrease R (number of red balls in the urn), how is the probability of drawing a red ball changing? Notice how the distribution (blue graph) becomes left- or right-skewed.

==Recognize that the sum of the probability of an event and the probability of its complement is equal to one ==

==Make approximate predictions using theoretical probability and proportions ==

==Collect and interpret data to show that as the number of trials increases, experimental probability approaches the theoretical probability==

* [[SOCR_EduMaterials_Activities_ScatterChart#Example_.28Human_Heights_and_Weights.29 | Scatter Plots of Human heights and weights]]

* [[AP_Statistics_Curriculum_2007_EDA_Pics#Dot_Plots | Dot Plots]]

* [[SOCR_EduMaterials_Activities_PieChart#Example_.28Lake_contamination_and_Mercury_in_Fish.29 | Pie Charts of Lake Contamination and Mercury in Fish]]

==Compare two similar sets of data on the same graph ==

* [[SOCR_EduMaterials_Activities_LineChart#Example_.28Human_Heights_and_Weights.29 | See this example of Line Plots for human Weight and Height]]

==Compare two different kinds of graphs representing the same set of data ==

* Compare the [[SOCR_EduMaterials_Activities_LineChart#Example_.28Human_Heights_and_Weights.29 | Line Plots]] and the [[SOCR_EduMaterials_Activities_ScatterChart#Example_.28Human_Heights_and_Weights.29 | Scatter Plots]] for the [[SOCR_Data_Dinov_020108_HeightsWeights | Human heights and weights dataset]].

==Propose and justify inferences and predictions based on data ==
Refer to the [[SOCR_EduMaterials_Activities_LineChart#Example_.28Human_Heights_and_Weights.29 | Line Plots]] and the [[SOCR_EduMaterials_Activities_ScatterChart#Example_.28Human_Heights_and_Weights.29 | Scatter Plots]] for the [[SOCR_Data_Dinov_020108_HeightsWeights | Human heights and weights dataset]].

* Is it likely that someone is 80 inches tall if his/her weight is only 100 pounds?
* What is a reasonable height in inches for someone who is 130 pounds in weight (notice that this is the mean weight of an individual, so we may expect the mean height (68.4 in) to be a reasonable estimate)?

==References==
* [http://www.schools.utah.gov/curr/core/corepdf/Mth7-12.pdf Utah Secondary Core Curriculum Standards for Statistics]
* [[EBook | Interactive Statistics Education EBook]]
* SOCR Home page: http://www.socr.ucla.edu

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 ===Solve counting problems using the Fundamental Counting Principle===
-Suppose that a task involves a sequence of '''k''' choices.  Let n<sub>1</sub> be the number of ways the 1<sup>st</sup> stage or event can occur and n<sub>2</sub> be the number of ways the 2<sup>nd</sup> stage or event can occur after the first stage has occurred.  Continuing in this way, let  be the number of ways the '''k'''<sup>nd</sup> stage or event can occur after the first '''k - 1''' stages or events have occurred.  Then the total number of different ways the task can occur is: <math>n_1 \times n_2 \times n_3 \times \cdots \times n_k</math>.
+Suppose that a task involves a sequence of '''k''' choices.  Let <math>n_1</math> be the number of ways the 1<sup>st</sup> stage or event can occur and <math>n_2</math> be the number of ways the 2<sup>nd</sup> stage or event can occur after the first stage has occurred.  Continuing in this way, let  be the number of ways the '''k'''<sup>nd</sup> stage or event can occur after the first '''k - 1''' stages or events have occurred.  Then the total number of different ways the task can occur is: <math>n_1 \times n_2 \times n_3 \times \cdots \times n_k</math>.
 * Example:  A deli has a lunch special which consists of a sandwich, soup, dessert and drink for $4.99.  If they offer the following choices, ''how many lunch specials are possible''?

← Older revision		Revision as of 19:07, 8 February 2008
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	===Solve counting problems using the Fundamental Counting Principle===		===Solve counting problems using the Fundamental Counting Principle===
		+
		+	Suppose that a task involves a sequence of '''k''' choices. Let n<sub>1</sub> be the number of ways the 1<sup>st</sup> stage or event can occur and n<sub>2</sub> be the number of ways the 2<sup>nd</sup> stage or event can occur after the first stage has occurred. Continuing in this way, let be the number of ways the '''k'''<sup>nd</sup> stage or event can occur after the first '''k - 1''' stages or events have occurred. Then the total number of different ways the task can occur is: <math>n_1 \times n_2 \times n_3 \times \cdots \times n_k</math>.
		+
		+	* Example: A deli has a lunch special which consists of a sandwich, soup, dessert and drink for $4.99. If they offer the following choices, ''how many lunch specials are possible''?
		+	** Sandwich: chicken salad, ham, and tuna, and roast beef
		+	** Soup: tomato, chicken noodle, vegetable
		+	** Dessert: cookie and pie
		+	** Drink: tea, coffee, coke, diet coke and sprite
		+
		+	We can use the basic counting principle:
		+
		+	There are '''k=4''' stages or events: ''choosing a sandwich, choosing a soup, choosing a dessert and choosing a drink''. There are ''4 choices for the sandwich, 3 choices for the soup, 2 choices for the dessert and 5 choices for the drink''. Putting that all together we get:
		+
		+	'''Number of choices = 4 (sandwiches) x 3 (soups) x 2 (desserts) x 5 (drinks) = 120'''. So there are exactly 120 lunch specials possible at this deli.

	===Calculate the probability of an event or sequence of events with and without replacement using models===		===Calculate the probability of an event or sequence of events with and without replacement using models===

@@ Line 1: / Line 1: @@
 == [[SOCR_Events_May2008 | SOCR May 2008 Event]] - Calculate probabilities of events and compare theoretical and experimental probability==
-==Solve counting problems using the Fundamental Counting Principle==
+===Solve counting problems using the Fundamental Counting Principle===
-==Calculate the probability of an event or sequence of events with and without replacement using models ==
+===Calculate the probability of an event or sequence of events with and without replacement using models===
 The [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html Ball and Urn Experiment] demonstrates the effects of sampling with or without replacement. Suppose we '''choose n balls''' at random from an urn containing '''R are red''' and ('''N-R) are green''' balls. For every trial, the '''number of red balls drawn (Y)''' are recorded numerically in the distribution table (on the right) and graphically in the distribution graph (blue). At each trial, the empirical frequency of Y is displayed in '''red''' the distribution graph. The [[SOCR_EduMaterials_Activities_BallAndRunExperiment | experimenter has the abilities to manipulate trials by choosing with replacement or without replacement in the list box and varying parameters N, R, and n with scroll bars]].
-[[Image:SOCR_Activities_BallAndUrnExperiment_Dinov_020808_Fig1.jpg|500px]]
+<center>[[Image:SOCR_Activities_BallAndUrnExperiment_Dinov_020808_Fig1.jpg|500px]]</center>
 * Why is there a difference in sampling balls from the urn with or without replacement?
@@ Line 12: / Line 12: @@
 * If we increase or decrease R (number of red balls in the urn), how is the probability of drawing a red ball changing? Notice how the distribution (blue graph) becomes left- or right-skewed.
-==Recognize that the sum of the probability of an event and the probability of its complement is equal to one ==
+===Make approximate predictions using theoretical probability and proportions===
-==Make approximate predictions using theoretical probability and proportions ==
+<center>[[Image:SOCR_Activities_BallAndUrnExperiment_Dinov_020808_Fig2.jpg|500px]]</center>
-==Collect and interpret data to show that as the number of trials increases, experimental probability approaches the theoretical probability==
+===Collect and interpret data to show that as the number of trials increases, experimental probability approaches the theoretical probability===
+Increase the number of experiments to 1,000 and click the '''Run''' button. Notice hos the empirical peobabilities (red graph) quickly approach the theioretical probabilities (blue graph) and similarrly the theoretical and empirical probabilities become numerically similar (see bottom-right table).
+<center>[[Image:SOCR_Activities_BallAndUrnExperiment_Dinov_020808_Fig3.jpg|500px]]</center>