SOCR EduMaterials Activities CardsCoinsSampling

Summary

This is a heterogeneous Activity that demonstrates different settings for sampling drawing synergistic parallels between empirical and theoretical probability calculations.

Exercises

• Exercise 1: Suppose 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. You can see the Standard 52-card deck here. Using the SOCR Card-Experiment perform 20 experiments and determine the proportion of the outcomes that event A has occurred. This would correspond to the empirical probability (the chance) of the event A. How close are the observed and the theoretical probabilities for the event A? Would the discrepancy between these increase or decrease if the number of hands drawn increases? First, do the experiment, and then report you findings!

• Exercise 2: Now we'll learn how to use these Card experimental results to compute expectations, summarize the outcomes of a large number of experiments, and validate these random simulations with the corresponding theoretical probabilities.
  • First, select the Card-Experiment. Then, we can run 10 (or more) experiments by clicking the <RUN> Button (>>). The output may look like this (recall that the actual cards will be different each time you perform the experiment):
  • Now select all outcome values in the eleven columns (recall that the first column is the index of the experiment, the remaining 2xN, in this case 10, columns represent the denomination (1<=Y<=13) and the suit (0<=Z<=3) of the cards in the randomly drawn hand). Save this selection (sky-blue highlight) in your mouse buffer, <CNTR>+C, or <APPLE>+C. Open a new browser tab or window, invoke the SOCR Charts, and select the SOCR Area-Chart Demo:
  • Press the <CLEAR> Button to remove the default data and go to the <DATA> tab where you will click on the Bottom-Right (empty) cell and hit <TAB> key to open a new 11-th column. Finally, click on the top-left cell and press the <PASTE> button to fill in the Card-Experiment results into the data-table of the Area-Chart
  • You will need to go to the <MAPPING> tab and map the Index column (1) onto Series, and the rest of the columns should be mapped to Categories. Click on <Update_Chart> button and select the <GRAPH> tab to see the visual depiction of the summary statistics for your results of the Card-Experiment.
  • The final result should look like this, where you can clearly identify the proportions, frequencies, and distributions of all the denominations and suits of the randomly drawn hands of cards that you did in the Card-Experiment above.

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• Exercise 3: 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. Under these fair coin assumptions, what is the (theoretical) probability that only 1 Head is observed in 6 tosses? Use the Binomial Coin Experiment to:
  • Empirically 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);
  • Empirically 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 empirical 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!

• Exercise 4: In the SOCR 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?
  • 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.
  • 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 obtained 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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• SOCR Home page: http://www.socr.ucla.edu

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