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Fundamental Counting Principle: If an event M can occur m ways and is followed by an event N that can occur n ways, then event M followed by event N can occur m n ways. Permutation: An arrangement of items in a particular order. The number of permutations of n items of a set arranged r items at a time is n Pr n! (n r )! n! n (n 1) ... 3 2 1 Combination: A selection in which order does not matter. Example: A club has nine members. In how many ways can a president, vice president, and secretary be chosen from its members of this club? Example: From 20 raffle tickets in a hat, four tickets are to be selected in order. The holder of the first ticket wins a car, the second a motorcycle, the third a bicycle and the fourth a skateboard. In how many different ways can these prizes be awarded? COMBINATIONS: ORDER DOES NOT MATTER! The number of combinations of n objects taken r at a time n! C is: n r r!(n r )! Example: 5 C3 Example: A club has nine members. In how many ways can a committee of three be chosen from the members of this club? Example: From 20 raffle tickets in a hat, four tickets are to be chosen at random. The holders of the winning tickets are to be awarded free trips to the Bahamas. In how many ways can the four winners be chosen? A pizza parlor offers the basic cheese pizza and a choice of 16 toppings. How many different kinds of pizza can be ordered at this pizza parlor? Guidelines for using Permutations and Combinations: - When we want to find the number of ways of picking r objects from n objects we need to ask ourselves: Does the order of the objects matter? - If the order matters, we use permutations - If the order doesn’t matter, we use combinations. Example: A chemistry teacher divides his class into eight groups. Each group submits one drawing of the molecular structure of water. He will select four of the drawings to display. In how many different ways can he select the drawings? Example: You will draw winners from a total of 25 tickets in a raffle. The first ticket wins $100. The second ticket wins $50. The third ticket wins $10. In how many different ways can you draw the three winning tickets? Example: How many 5 card hands can be dealt from a deck of 52 cards? Example: A committee of seven consisting of a chairman, a vice chairman, a secretary, and four other members, is to be chosen from a class of 20 students. In how many ways can this committee be chosen? Probability: Experimental Probability of an event: Example: Of the 60 vehicles in a teacher’s parking lot today, 15 are pickup trucks. What is the experimental probability that a vehicle in the lot is a pickup truck? Example: A class tossed coins and recorded 161 heads and 179 tails. What is the experimental probability of heads? Of tails? Theoretical Probability: - Sample Space: the set of all possible outcomes to an experiment or activity. - Equally Likely: When each outcome in a sample space has the same chance of occurring. - If a sample space has n equally likely outcomes and an event A occurs in m of these outcomes, then the theoretical probability of event A is: Example: What is the theoretical probability of each event? a. Getting a 5 on one roll of a standard number cube b. Getting a sum of 5 on one roll of two standard number cubes Using Combinations to find probability: Example: A five-card poker hand is drawn from a standard deck of 52 cards. What is the probability that all five cards are spades? Example: A bag contains 20 Frisbees, of which four are defective. If two Frisbees are selected at random from the bag, what is the probability that both are defective? Complement of an Event: The complement of an event E is the set of outcomes in the sample space that is not in E. We denote the complement of an event E by E’. Probability of the complement of an event: Example: An urn contains 10 red marbles and 15 blue marbles. Six marbles are drawn at random from the urn. What is the probability that at least one marble is red?