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Chapter 8A: Probability. A phenomenon or trial is said to be random if individual outcomes are uncertain. The probability of any outcome of a random event is the proportion of times the out come would occur in a very long series of repetitions. The sample space of a random phenomenon is the set of all possible outcomes that cannot be broken down into simpler components. An event is any outcome or any set of outcomes of a random phenomenon. A probability model is a mathematical description of a random phenomenon consisting of two parts: a sample space and a way of assigning probabilities to events. There are a few rules in probability. The ones we are going to focus on are as follows: 1. The probability of an event to happen must be between 0 and 1 (or 0% to 100%). An event with probability of zero is deemed as impossible, while an event with probability of one is deemed as a certainty. 2. If the probability of an event to happen is P (A), then the probability of that event to not happen is 1 − P (A). 3. If two events A and B are disjoint, then the probability of A or B happening is P (A) + P (B). 4. If two events A and B are not disjoint, then the probability of A or B happening is P (A) + P (B) minus the probability of both things happening. 5. All possible outcomes together must add up to 1 (or 100%). Example 1: Coin Tossing. A fair die is a die in which all six sides are equally likely to come show up. Answer the following: 1. 2. 3. 4. What What What What is is is is the the the the sample space if you were to toss the die once? probability that you roll a 7? probability that you roll an even number? probability that you roll less than 6? Example 2: Benford’s Law. Benford’s law, also called the first-digit law, states that in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed in a specific, non-uniform way. This peculiar phenomenon happens very often in naturally occurring data sets, and an explanation of this can be found on p. 256. By not allowing the first digit to be zero, we have the following probability model: First Digit Probability 1 0.301 2 0.176 3 0.125 4 0.097 5 0.079 6 0.067 7 0.058 8 0.051 9 0.046 Answer the following, using the table above: 1. 2. 3. 4. What What What What is is is is the the the the probability probability probability probability of of of of having having having having a first digit of a one or a two? a first digit that is not a seven? an even first digit? a multiple of three as a first digit? Section 8B: Combinatorics. Combinatorics is the study of methods of counting. We start by definition: The Fundamental Principle of Counting states that if you have a ways of choosing one thing, b ways of choosing a second after the first is chosen, etc..., then the total number of choice sequences is a · b · . . . . Example 3: Choosing Toppings. You are at a summer barbecue and you are dressing up your burger. At the condiment table, you see mustard, mayonnaise, ketchup, and sweet relish all in a squeezable container. Also on the table, you also see sliced tomatoes, sandwich pickles, and lettuce. How many possible ways can you dress up your burger? There are three main combinatorics principle, which will be illustrated in the following examples. Example 4: DNA Codons. DNA consists of a long sequence of of the nucleotides adenine, cytosine, guanine, and thymine (abbreviated as A, C, G, T). A codon is a string of three consecutive nucleotides. How many possible codons can be created with these four nucleotides? Example 5: Beauty Pageant. Alice, Beth, Christie, Danielle, and Esther are the final five contestants in a beauty pageant. One will be named second runner up, one will be named first runner up, and one will be named the winner. How many possible ways can they be arranged do the three placements? Example 6: State Quarters. You have a four state quarters in your pocket, and the states represented are Tennessee, Rhode Island, Alabama, and Pennsylvania. How many possible combinations are possible if you pick out three of those four quarters from your pocket? • The first principle illustrated is an ordered set with replacement. In the example, we can keep reusing our choices for all the available slots. (AAA is allowed.) • The second principle illustrated is an ordered set without replacement, which is also called permutations. In this example, once we use up one choice, we cannot use it again. (AA is not allowed.) The shorthand for permutation is n Pr , with n being the number of items and r being the number of ordered slots available. • The third principle illustrated is an unordered set without replacement, which is also called combinations. In this case, a selection of TAR is the same as a selection of RAT, and ART. The shorthand for combination is n Cr , with n being the number of items and r being the number in the desired unordered set. For more on these topics, please see p. 258 - 260. Your calculator will be of great help here. You should have a combination and a permutation button in your calculator. However, it will be up to you to determine what the situation calls for. Try the following: 1. A bike lock has four digits, and only one four digit number will unlock the contraption. How many possible combinations are there? 2. In the Georgia Lottery’s Fantasy Five, five numbers are drawn at random from 39 possible numbers. Matching these five numbers in any order will mean you won the jackpot. If you cannot make up your mind on your five numbers, there is a Quick Pick option, in which the computer will randomly choose for you. How many possible five number sets can quick pick generate? 3. An NBA team has 12 players on its active roster, but only five players can start. There are generally five positions: point guard (PG), shooting guard (SG), small forward (SF), power forward (PF), and a center (C). Pretend for now that all 12 players can play at any starting position and all are equally likely to be chosen to start. How many possible ways can you create a starting lineup? 4. How many possible five digit zip codes do not contain the number zero? 5. A password is considered alphanumeric if it contains both letters (A to Z) and numbers (0 to 9). How many possible passwords can be created if it must be six characters long and repetition is not allowed? 6. The Miss America pageant has 53 contestants: one from each state, one from Washington, D.C., one from Puerto Rico, and one from the U.S. Virgin Islands. How many possible ways can ten finalists be selected out of the 53 contestants?