Probability

This week, we're finishing up counting and moving into probability.   Be careful when deciding what counting technique to use for a problem.  You should first see if you can break the problem into a sequence of tasks for which you can count the number of ways to perform each task.  Then, applying the Fundamental Counting Principle, you multiply the number of ways to do each task to get the total number of ways to do everything together.  When deciding whether to use a permutation or a combination to count, you should ask yourself "Does order matter?".  If the answer is yes, then you should use a permutation P(n, r).  If order doesn't matter, use a combination C(n, r).

Last time I taught this course, I had a student ask about counting cards.  The podcast This American Life has an interesting episode about card counting.  You can listen to it here.  Listen to the podcast and tell me what you learned in the comments.

Now, we're going to apply our counting skills to find probabilities.  On Tuesday, we talked about sample spaces and events.  Remember that the sample space is the list of all possible outcomes of a random experiment.  The probability of an event is the proportion of times that event is expected to occur in a long sequence of repetitions of a random process.  We looked at an app that models this for the probability of flipping heads when you flip one coin.  The Binomial Coin Experiment App allows you to change the number of coins being flipped and calculates the number of faces showing heads on each flip.  Click on the link and play around with the app.  Try flipping 3 coins (change to n=3 and leave p=0.50) and let the app run for 10000 flips.  The table to the right gives the probability of getting 0, 1, 2,  or 3 heads under the "Dist" column and also displays the empirical probabilities under the "Data" column.  Are these numbers close after 10,000 flips?  How did they compute those probabilities?