2.  Counting  Fundamental Counting principle  Factorials  Permutations and combinations  Probability  Complementary events  Compound events  Independent events  Dependent events

3.  Consider this:  You have an exam with five true or false questions. How many different ways are there to answer the test?  The fundamental counting principle states that: "If there are r ways to do one thing, and s ways to do another, and t ways to do a third thing, and so on..., then the number of ways of doing all those things at once is r * s * t etc...."

4.  You have four pairs of pants, five shirts, and six pairs of shoes. How many outfits can you make?  You are making license plates, but you are limited to letters in the first three spaces and numbers in the final three spaces. How many license plates could you make?

5.  A permutation is an ordering or arrangement.  Suppose that there are five students who need to ask for help on their pre-calculus homework. How many different orders could there be?  What if there were seven students instead of five?

6.  A more efficient way to find the products is the previous problems is factorial notation.  Factorial notation is defined as follows: For any natural number n, n! = n(n-1)(n- 2)…(2)(1) 0! = 1  Find 2!, 5!, and 5!  Use your calculator to find 13! And 20!

7.  There are 15 students in speech. How many arrangements are there for 3 students to give a speech?  P(n, r) represents the number of permutations for n elements, taken r at a time.  P (n, r) = n!/[(n-r)!] = n(n-1)(n-2) {for a total of r factors}  Try these:  P(7, 2)  P(10, 3)

8.  A combination is not a ordering or arrangement, but a subset of a set of elements.  Consider: you have 4 people in class ready to present, but you want to pick a team of two people to make a presentation. How many choices do you have?  C(n, r) represents this choice.  C(n, r) = P(n, r)/r! = n!/[(n-r)!*r!)  Try these:  C(10, 2)  C (30, 25)

9.  In permutations, order matters.  AB ≠ BA  In combinations, order does not matter.  AB = BA

10.  Probability is the likelihood that an event will occur.  It is between 0 and 1.  A probability of 0 means an event will never happen.  A probability of 1 means an event is certain to happen.  A few terms:  Outcome: the result from an experiment  Sample space: the set of all outcomes  Event: subset of sample space

11.  Probability is computed by dividing the favorable outcomes by possible outcomes. P(E) = n(E)/n(S) Where n(E) is the number of outcomes that are favorable and n(S) is the total outcomes  Find the probability of drawing a queen from a deck of cards.

12.  An event and its complement take up the entire sample space.  This is often shown as E and E’.  For example, when rolling dice, the probability of rolling a six and the probability of not rolling a six are complements. What is the probability of each?  P(E)+ P(E’) = 1

13.  This is the probability of two events occurring.  It is measured by the probability of each event occurring minus any overlap.  Try this one:  What is the probability of rolling two dice and getting either the sum of eight or rolling a pair?

14.  Suppose you toss a coin and then toss it again. What is the probability that you will get two heads?  This is the probability of independent events.  It is the probability of one event times the probability of the other.

15.  Imagine that you trying to find the probability of drawing two hearts from a deck of cards. In scenario 1, the card you have drawn is replaced. In scenario 2, the card is not replaced. What is the probability of each scenario?  Scenario 2 is the probability of dependent events.  This is the probability of the 1 st event and then the probability of the 2 nd event, assuming that the 1 st event happened.

16.  Pages 926 -928 #32-44 even, #56-68 even  Page 951 #22-30 even