Difference between revisions of "SOCR Events May2008 C3 S1"
(New page: == SOCR May 2008 Event - Calculate probabilities of events and compare theoretical and experimental probability== ==Solve counting problems using the Fundamental...) |
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== [[SOCR_Events_May2008 | SOCR May 2008 Event]] - Calculate probabilities of events and compare theoretical and experimental probability== | == [[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]]. | 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? | * Why is there a difference in sampling balls from the urn with or without replacement? | ||
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* 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. | * 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=== | ||
| + | For the '''Ball and Urn Experiment''' above, let E={Event that a sample of 2 balls contain at least one Red ball}. If N=50, R=25 and n=2 and we sample with replacement, what is the complement of this event, E<sup>c</sup>? That are the probabilities of E and E<sup>c</sup>? Does P(E) + P(E<sup>c</sup>)=1? Is this expected? | ||
| − | == | + | ===Make approximate predictions using theoretical probability and proportions=== |
| + | Suppose we did not know the values of R (real number of red balls in the urn) and we wanted to estimate R from a sample of 20 balls randomly chosen with replacement. How can we do that and what is the appropriate estimate for R? Note that each time you run this experiment, this estimate will slightly vary as this is a random experiment where chance plays role. | ||
| − | + | <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). | |
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Revision as of 13:56, 8 February 2008
Contents
- 1 SOCR May 2008 Event - Calculate probabilities of events and compare theoretical and experimental probability
- 1.1 Solve counting problems using the Fundamental Counting Principle
- 1.2 Calculate the probability of an event or sequence of events with and without replacement using models
- 1.3 Recognize that the sum of the probability of an event and the probability of its complement is equal to one
- 1.4 Make approximate predictions using theoretical probability and proportions
- 1.5 Collect and interpret data to show that as the number of trials increases, experimental probability approaches the theoretical probability
- 2 References
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 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 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.
- 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
For the Ball and Urn Experiment above, let E={Event that a sample of 2 balls contain at least one Red ball}. If N=50, R=25 and n=2 and we sample with replacement, what is the complement of this event, Ec? That are the probabilities of E and Ec? Does P(E) + P(Ec)=1? Is this expected?
Make approximate predictions using theoretical probability and proportions
Suppose we did not know the values of R (real number of red balls in the urn) and we wanted to estimate R from a sample of 20 balls randomly chosen with replacement. How can we do that and what is the appropriate estimate for R? Note that each time you run this experiment, this estimate will slightly vary as this is a random experiment where chance plays role.
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).
References
- Utah Secondary Core Curriculum Standards for Statistics
- Interactive Statistics Education EBook
- SOCR Home page: http://www.socr.ucla.edu
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