Difference between revisions of "SOCR EduMaterials Activities DieCoin"

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(→‎This is an activity to explore the bivariate distribution of X and Y where X is the number shown when a die is rolled and Y is the number of heads when a coin is tossed number of times equal to X.)
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== This is an activity to explore the bivariate distribution of X and Y where X is the number shown when a die is rolled and Y is the number of heads when a coin is tossed number of times equal to X. ==
 
== This is an activity to explore the bivariate distribution of X and Y where X is the number shown when a die is rolled and Y is the number of heads when a coin is tossed number of times equal to X. ==
  
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* "'Description"':
 
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* '''Exercise 1''': Supose a random card is drawn from a standard well-shuffled deck of 52 playing cards.  What is the theoretical probability of the event A = {the drawn card is a king or a club}? In this applet an outcome of a King is recorded in the first variable Y and corresponds to Y=13; the second variable Z corresponds to the suit of the card with a Club, ♣, represented by Z=0.  Using the SOCR [http://socr.stat.ucla.edu/htmls/SOCR_Experiments.html Card-Experiment] perform 20 experiments and determine the proportion of the outcomes the event A was observed. This would correspond to the emperical probability (the chance) of the event A. How close are the '''observed''' and the '''theoretical''' proabilities for the event A? Would the discrepancy between these increase or dicrease if the number of hands drawn increases? First do the xperiment and then report you findings!
 
  
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<center>[[Image:SOCR_Activities_CardCoinSampling_Dinov_092206_Fig1.jpg|300px]]</center>
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* '''Description''': A die is rolled and the number observed X is recorded.  Then a coin is tossed number of times equal to the value of X.  For example if X=2 then the coin is tossed twice, etc.  Let Y be the number of heads observed.  Note:  Assume that the die and the coin are fair.
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* '''Exercise 2''': Suppose we need to validate that a coin given to us is '''fair'''. We toss the coin 6 times independently and observe only one Head. If the coin was fair P(Head)=P(Tail)=0.5 and we would expect about 3 Heads and 3 Tails, right? Under these fair coin assumptions what is the ('''theoretical''') probability that only 1 Head is observed in 6 tosses? Use the [http://socr.stat.ucla.edu/htmls/SOCR_Experiments.html Binomial Coin Experiment] to:
 
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** Emprerically compute the odds (chances) of observing one Head in 6 fair-coin-tosses (run 100 experiments and record the number of them that contain exactly 1 Head);
 
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** Emperically estimate the Bias of the coin we have tested. Experiment with tossing 30 coins at a time. You should change the p-value=P(Head), run experiments and pick a value on the X-axis that the emperical distribution (red-histogram) peaks at. Perhaps you want this peak X value to be close to the observed 1-out-of-6 Head-count for the original test of the coin. Explain your findings!
 
  
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<center>[[Image:SOCR_Activities_CardCoinSampling_Dinov_092206_Fig2.jpg|300px]]</center>
 
  
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* '''Exercise 3''': In the SOCR [http://socr.stat.ucla.edu/htmls/SOCR_Experiments.html Ball and Urn Experiment] how does the distribution of the number of red balls (Y) depend on the sampling strategy (with or without replacement)? Do N, R and n also play roles?
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* '''Exercise 1''': Construct the joint probability distribution of X and Y.
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** Suppose N=100, n=5, R=30 and you run 1,000 experiments. What proportion of the 1,000 samples had zero or one red balls in them? Record this value.
 
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** Now run the Binomial Coin Experiment with n=5 and p= 0.3. Run the Binomial experiment 1,000 times? What is the proportion of observations that have zero or one head in them? Record this value also. How close is the proportion value you abtained before to this sample proportion value? Is there a reason to expect that these two quantities (coming from two distinct experiments and two different underlying probability models) should be similar? Explain.
 
  
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<center>[[Image:SOCR_Activities_CardCoinSampling_Dinov_092206_Fig3.jpg|300px]]</center>
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* '''Exercise 2''': Find the conditional expected value of Y given X=5.
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* '''Exercise 3''': Find the conditional variance of Y given X=5.
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* '''Exercise 4''':  Find the expected value of Y.
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* '''Exercise 5''': Find the standard deviation of Y.
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* '''Exercise 6''': Graph the probability distribution of Y.
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* '''Exercise 7''': Use SOCR [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html Experiments] and choose "Die Coin Experiment" to graph and print the empirical distribution of Y when the experiment is performed:
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a. n = 1000 times.
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b. n= 10000 times
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<center>[[Image:SOCR_Activities_DieCoinExperiment_Christou_092206_Fig1.jpg |600px]]</center>
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* '''Exercise 8''': Compare the theoretical mean and standard deviation of Y (exercise 4 and 5) with the empirical mean and standard deviation found in exercise 7.
  
 
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* SOCR Home page: http://www.socr.ucla.edu
 
* SOCR Home page: http://www.socr.ucla.edu
  
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{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_ConfidenceIntervals}}
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{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_DieCoin}}

Latest revision as of 10:46, 17 September 2007

SOCR Educational Materials - Activities - SOCR Die Coin Activity

This is an activity to explore the bivariate distribution of X and Y where X is the number shown when a die is rolled and Y is the number of heads when a coin is tossed number of times equal to X.

  • Description: A die is rolled and the number observed X is recorded. Then a coin is tossed number of times equal to the value of X. For example if X=2 then the coin is tossed twice, etc. Let Y be the number of heads observed. Note: Assume that the die and the coin are fair.


  • Exercise 1: Construct the joint probability distribution of X and Y.


  • Exercise 2: Find the conditional expected value of Y given X=5.


  • Exercise 3: Find the conditional variance of Y given X=5.


  • Exercise 4: Find the expected value of Y.


  • Exercise 5: Find the standard deviation of Y.


  • Exercise 6: Graph the probability distribution of Y.


  • Exercise 7: Use SOCR Experiments and choose "Die Coin Experiment" to graph and print the empirical distribution of Y when the experiment is performed:

a. n = 1000 times.

b. n= 10000 times

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  • Exercise 8: Compare the theoretical mean and standard deviation of Y (exercise 4 and 5) with the empirical mean and standard deviation found in exercise 7.

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