Introduction to Probability WebQuest

Why are we doing this?

What should happen? What did happen? How can I predict what will happen next? In the realm or mathematics, these are the basic questions that are the driving force behind the study of probability. In essence, there are two different kinds of probability: Theoretical Probability - making predictions on what should happen, and Experimental Probability - making predictions on what has already happened. To illustrate this difference, we will conduct a little experiment.

Technical Stuff:

On the Internet or in your book, find the definitions of the following terms. When done, you can click on the small triangles on the right side of each box to reveal the answers: Experiment The act of making an observation or taking a measurement, such as flipping a coin. Event Any subset of all the possible outcomes of an experiment. Theoretical Probability The process of predicting the likelihood of the next occurrence of an event based on the mathematical calculations. Experimental Probability The process of predicting the likelihood of the next occurrence of an event based on the results of an experiment. Also, recall the formulas for the area of a circle and the area of a rectangle. Ac = 3.14 × r 2 Ar = w × h

First, let's do the math.

Go to the Area Probability: (Throw Darts!) website. Once there, play with some of the controls to understand the various functions of the features. When ready, be sure to hide the "Show areas" and "Show area ratio" features, and proceed by answering the following: According to the settings displayed, what is the radius of the circle?   What is the area of the circle, rounded to the nearest hundredth (denote with Ac)?   According to the settings displayed, what are the width and height of the rectangle?   What is area of the rectangle, rounded to the nearest hundredth (denote with AR)?   What is the ratio of the areas, rounded to the nearest thousandth? That is, what is Ac/AR?   Once you have answered the above questions, reveal the "Show areas" and "Show area ratio" calculations to see if you did them correctly.   What is the significance of the ratio of the two areas?   Based upon these calculations, what is the probability of hitting the circle when a dart thrown?   Would you call this the theoretical or experimental probability? Why?

Now, let's throw some darts.

After you have answered the questions listed above, throw some darts but clicking on the "Throw 1000" buttons 10 times. Each time the darts are thrown, record the number and percent (write as a decimal, rounded to the nearest thousandth) of the darts that hit the circle. Record your data in a table similar to the following. Trial # # of Darts that Hit the Circle % of Darts that Hit the Circle 1     2     3     4     5     6     7     8     9     10       If you forget to record a particular set of data, don't worry. If you click on the leftmost clipboard found in the lower right-hand part of the screen, you can get to a record of your activities. Based on your 10 trials, what is the average (mean) of the percent of darts that hit the circle?   How does this average compare to the ratio of the two areas you found in the previous calculation?   Based upon this experiment, what is the probability of hitting the circle with the next dart thrown?   Would you call this the theoretical or experimental probability? Why?

Questions to think about.

To fully understand the results of this experiment, you should think about the following questions. Can a dart hit the circle, but not be within the rectangle? Answer. --> No. Since the circle is contained within the rectangle, any dart that hits the circle HAS to hit the rectangle.   Can you adjust the radius of the circle so that the area of the circle is greater than the area of the rectangle? Why or why not? Answer. --> No. For starters, the program won't let this be done. Why? Since any dart that hits the circle must also hit the rectangle, the circle must be completely enclosed within the rectangle. Therefore, the area of the circle must always be less than the area of the rectangle. By the way, they can't be equal, either.   If you click and hold the circle, it can be dragged to different places within the rectangle. Will this movement change the results of the experiment? Answer. --> Nope. Moving the circle does not change its area. As long as the areas are unchanged, the ratio of the areas also remains unchanged.