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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.
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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)
