Counting Techniques & Probability
220 likes | 693 Views
Counting Techniques & Probability. Dr. Johnny Duke Georgia Highlands College Spring 2007. Counting Techniques. Fundamental Counting Rule (aka The Multiplication Rule) Permutations Combinations. Decision Flow Chart. Is repetition allowed? Yes, Use fundamental counting rule.
Counting Techniques & Probability
E N D
Presentation Transcript
Counting Techniques & Probability Dr. Johnny Duke Georgia Highlands College Spring 2007
Counting Techniques • Fundamental Counting Rule (aka The Multiplication Rule) • Permutations • Combinations
Decision Flow Chart • Is repetition allowed? • Yes, Use fundamental counting rule. • No, Go to second question. • Is order important? • Yes, Use permutation. • No, Use combination.
Fundamental Counting Rule • A sequence of n events each having ki possible outcomes has total outcomes equal to k1 x k2 x k3 x …. x kn Tree Diagrams
Fundamental Counting Rule Bob has a red, a blue, and a green shirt. He has tan and black pants. How many different outfits does Bob have? Tan Black Red Tan Blue Black Tan Green Black x 3 2
Permutations • Repetition is not allowed & order is important • Note:
Combinations • Repetition is not allowed and order is not important • This is the number of subsets of a set.
Enter n, hit MATH, arrow to PRB, choose nPr, enter r, hit enter. Enter n, hit MATH, arrow to PRB, choose nCr, enter r, hit enter Permutations & Combinations on the TI 83
Probability Terminology • Probability Experiment: Toss a coin twice and record the results. • Simple outcome: HH or HT • Sample Space: The set of all simple outcomes. In this case—HH, HT, TH, TT • Event: Exactly one head, exactly two heads, at least one head, etc. • Certain event: Probability is 1—at least zero heads. • Impossible event: Probability is 0—exactly three heads.
Probability Assignments • Classical • Assume that all simple outcomes are equally likely to occur. • Empirical • Probability assignments are based on relative frequencies of occurances • Subjective • Probability is based on personal experience (without a scientific study) and/or a hunch
Addition Rule • Mutually exclusive events: • Events that cannot happen at the same time. • Example: Get exactly two heads, get exactly one head. • P(A or B) = P(A) + P(B) – P(A and B)
Multiplication Rule • Independent Events: • The probability of event A has no effect on the probability of event B occurring. • If independent, then P(A and B)= P(A) x P(B) • If dependent, then P(A and B) = P(A) x P(B|A)
