1. June 4, 2009 Dr. Lisa Green

2.  Main goal: Understand the difference between probability and statistics.  Also will see: Binomial Model Law of Large Numbers Monte Carlo Simulation Confidence Intervals

3. ModelData Probability Statistics Model: An idealized version of how the world works. Data: Collected observations.

4.  Probability: The model is known, and we use this knowledge to describe what the data will look like.  Statistics: The model is (partially) unknown, and we use the data to make conclusions about the model.

5.  There are repeated trials, each of which has only two outcomes. (Success or Failure)  The trials are independent of each other.  The number of trials (n) is known.  The probability of success on each trial (p) is constant.

6.  Flip a coin 10 times, count the number of heads seen. n=10, p=0.50  Test 100 newly manufactured widgets, count the number that fail to work. n=100, p=?  Give a blood test to 35 volunteers, count the number with high cholesterol. n=35, p=?

7.  Pick a point at random inside the unit square.  If it is also inside the arc of the unit circle, count it as a success. If not, count it as a failure.  What is the probability of a success? 1 unit

8.  We know that the probability of success is π/4.  If we repeat this trial n times, we have a binomial experiment.  If n=100, we expect between 71 and 86 of the trials to end up successes. (95% of the time)

9. nLower boundUpper bound 1007186 1000760810 1000077747934 1000007828678794 1000000784594786202 10000000 78514387856526 7851438/10000000 * 4 = 3.1406 and 7856526/10000000 * 4 = 3.1426 This is the law of large numbers in action. If we didn’t already know the value of pi, and we had a lot of time, we could use this to estimate pi. Using random processes to estimate constant numbers is called Monte Carlo Simulation. A simulation of this is at http://polymer.bu.edu/java/java/montepi/montepiapplet.html http://polymer.bu.edu/java/java/montepi/montepiapplet.html

10.  We knew the model.  We knew the values of all constants.  We used that knowledge to make predictions about what was going to happen.

11.  Ask a randomly chosen person whether they know anyone affected by layoffs at GM.  If the response is yes, count this as a success. If not, count it as a failure.  What is the probability of a success?

12.  We don’t know the probability of success. Let’s call it p for now.  If we repeat the trial n times, and are careful about which people we talk to, we have a binomial experiment.  If we talk to 100 people, and 17 say they know someone affected by layoffs at GM, then the value of p is somewhere between 0.096 and 0.244 (95% confidence).

13. nObserved successes Lower BoundUpper Bound 100170.0960.244 10001700.1470.193 1000017000.1630.177 100000170000.1680.172 10000001700000.1690.171 Note: There are obviously logistical difficulties in asking a million people a question. Confidence intervals have confidence levels. The ones above are at the 95% confidence level. Here is an applet that lets you explore what the confidence level means: http://www.rossmanchance.com/applets/Confsim/Confsim.html http://www.rossmanchance.com/applets/Confsim/Confsim.html

14.  We knew the model, but not the value of all constants.  We used observed data to tell us something about the model (the unknown constant).

15.  Buffon’s Needle http://www.mste.uiuc.edu/reese/buffon/buffon. html http://www.mste.uiuc.edu/reese/buffon/buffon. html  Reese’s Pieces Applet http://www.rossmanchance.com/applets/Reeses /ReesesPieces.html http://www.rossmanchance.com/applets/Reeses /ReesesPieces.html  CAUSEweb http://www.causeweb.org/http://www.causeweb.org/

17. N=10, p=0.14 N=100, p=0.14