1. Ch2.3-2.4 2.3 Counting Techniques Product Rule If the first element or object of an ordered pair can be used in n 1 ways, and for each of these n1 ways the second can be selected n 2 ways, then the number of pairs is n 1 n 2. **Note that this generalizes to k elements (k – tuples) Permutations Any ordered sequence of k objects taken from a set of n distinct objects is called a permutation of size k of the objects. Notation: P k,n

2. Ch2.3-2.4 Combinations Given a set of n distinct objects, any unordered subset of size k of the objects is called a combination. Notation:

3. Ch2.3-2.4 2.4 Conditional Probability For any two events A and B with P(B) > 0, the conditional probability of A given that B has occurred is defined by Which can be written:

4. Ch2.3-2.4 2.4 The Law of Total Probability Let the events A 1, A 2,…, A k be mutually exclusive and exhaustive events. The for any other event B,

5. Ch2.3-2.4 2.4 Bayes’ Theorem Let A 1, A 2, …, A n be a collection of k mutually exclusive and exhaustive events with P(A i ) > 0 for i = 1, 2,…,k. Then for any other event B for which P(B) > 0 given by

6. Ch2.3-2.4 2.4 Example 3 A blood test detects a certain disease 99% of the time when the disease is present. When a healthy person is tested, however, there is a 2% that the test will say he or she has the disease. Suppose 0.5% of the population has the disease. Find the conditional probability that a randomly tested person has the disease given his or her test says that he or she has it.

7. Ch2.3-2.4 2.4 Example 4 Three different machines M 1, M 2, M 3 are used to make a large batch of similar items. Suppose 20% of the items are produced by M 1, 30% by M 2, 50% by M 3. Suppose also that 1% of the items produced by M 1 are defective, as are 2% of those produced by M 2 and 3% of those produced by M 3. If one item is selected at random from the entire batch and is found to be defective, what is the probability that it was produced by M 2 ?