Part 1: Expected Outcomes with Quarters

(1) Suppose a coin is flipped one time. What is the sample space of this event? How many ways could a head (success) occur?

(2) Suppose a coin is flipped ten times. Predict how many heads will result. Justify your predictions.

(3) Conduct this experiment and record results. Did the actual outcome match your prediction? Compare your results with others in the class.

(4) Explain the difference between experimental and theoretical probabilities. How does this relate to your prediction and the actual outcome of the experiment?

Repeat task 3, but spin the quarter instead of flipping it. Compare the results of the two methods.  Explain any differences in the results.

(5) Can the following probabilities be determined theoretically and/or experimentally? Give your reasoning for each. 
a.  The probability that a '3' will be rolled on a fair die.  
b.  The probability that a certain person will develop heart disease.  
c.  The probability that the spinner below will land on the blue.  

d. The probability that flipping ten coins will result in exactly five heads.

This can be done experimentally by analyzing the results of the class or theoretically by utilizing binomial probability formulas.

Part 2: Investigating Probabilities Interactively

Go to the "Area Probability (Throw Darts!)" Activity on ExploreMath.  When the activity loads it will look like the screen shot below.

(6) Using the sliders, set the width and height to 10. Now set the radius of the circle to 2.

(7) Predict the percentage of darts that will land in the circular region if 100 darts are thrown. Select the 'Throw 100' button. Was your prediction close to the actual result? Repeat the throwing process several times. Was your prediction close to the average?

Select the 'Calculate data values' clipboard to see a running tally of the results.

(8) Select the 'Show areas' button. How do these areas relate to the number of darts hitting the circle? Deselect the 'Show areas' button.

(9)   Predict the percentage of darts that will land in the circular region if the radius of the circular region is doubled and 100 darts are thrown. How did you determine your prediction?

(10) Move the radius slider to 4 and look at the figures. Do the visual images match your prediction? Select the 'Show areas' button. Do these numbers match your prediction? If not, why not?

(11)  Select the 'Throw 100' button.  Was your prediction close to the actual result? Repeat the throwing process several times.  Was your prediction close to the average percentage of these trials?

(12)  How do A_c and A_r relate to the number of darts hitting the circle? How do these areas compare to the areas when the circle had a radius of 2?

(13)   Find dimensions of the rectangle and circle that will result in a high likelihood of exactly 20 out of 100 darts landing in the circular region.  Use the sliders to adjust the figures to these dimensions.  Explain why you chose these dimensions.  Run fifty trials.  Using the clipboard, compare the mean of the fifty trials to 0.20.

(14) Examine the distribution of percentages. What would a histogram depicting this distribution look like? Which interval of the histogram would have the greatest frequency? What could be said about the frequencies in intervals 'A' and 'K'?

Chart 1

F r e q u e n c y
	4-6     7-9     10-12   13-15  16-18  19-21   22-24  25-27  28-30  31-33  34-36

(15)  Using Chart 1, construct a histogram of the actual results.

Part 3: Finding an Experimental Pi

(16)  Move the width and height sliders to 10. Now move the radius sliders to 5. The circle is now inscribed in the square.

(17)  Run twenty trials of throwing 1000 darts. Select the 'Calculate data values' clipboard. According to the table, what was the mean percentage of darts that landed in the circular region? Does this number look familiar?

(18)  Multiply the mean by 4. Does the product look familiar? Why?

(19)  Write the ratio of the areas in algebraic form and simplify. What is the result?

Extensions:  

http://www4.ncsu.edu/~hsdrier/PETourpage1.html