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Outcomes, Counting, Countability, Measures and Probability OPRE 7310 Lecture Notes by Metin C¸Akanyıldırım Compiled at 02:01 on Thursday 20Th August, 2020

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Outcomes, Counting, Countability, Measures and Probability OPRE 7310 Lecture Notes by Metin C¸Akanyıldırım Compiled at 02:01 on Thursday 20Th August, 2020 Outcomes, Counting, Countability, Measures and Probability OPRE 7310 Lecture Notes by Metin C¸akanyıldırım Compiled at 02:01 on Thursday 20th August, 2020 1 Why Probability? We have limited information about the experiments so we cannot know their outcomes with certainty. More information can be collected, if doing so is profitable, to reduce uncertainty. But some amount of uncertainty always remains as information collection is costly and might even be impossible or inaccurate. So we are often bound to work with probability models. Example: Instead of forecasting the number of probability books sold at the UTD bookstore tomorrow, let us ask everybody (students, faculty, staff, residents of Plano and Richardson) if they plan to buy a book tomor- row. Surveying potential customers in this manner is always possible. But surveys are costly and inaccurate. Before setting up probability models, we observe experiments and their outcomes in real-life to have a sense of what is likely to happen. Deriving inferences from observations is the field of Statistics. These inferences about the likelihood of outcomes become the ingredients of probability models that are designed to mimic the real-life experiments. Probability models can later be used to make decisions to manage the real-life contexts. Example: Some real-life experiments that are worthy of probability models are subatomic particle collisions, genetic breeding, weather forecasting, financial securities, queues. 2 An Event – A Collection of the Outcomes of an Experiment Outcome of an experiment may be uncertain before the experiment happens. That is, the outcome may not be determined with (sufficient) certainty ex ante. Here the word experiment has a broad meaning that covers more than laboratory experiments or on-site experiments. It covers any action or activity whose outcomes are of interest. This broader meaning is illustrated with the next example. Example: As an experiment, we can consider an update of Generally Accepted Accounting Principles (GAAP) issued by Financial Accounting Standards Board (FASB.org). Suppose that the board is investi- gating an update of reporting requirements for startups (more formally, Development Stage Entities). The board can decide to keep (K) the status quo, increase (I) the reporting requirements or decrease (D) them. Although accounting professionals can assess the likelihood of each of the outcomes K, I and D, they cannot be certain whether the board’s discussion will lead to I or D, or K, so the outcomes of the update experiment are uncertain. Sufficiency of certainty depends on the intended use of the associated probability model. A room ther- mostat may be assumed to be showing the room temperature with sufficient certainty for the purpose of controlling the temperature with an air conditioner. The same thermostat may have insufficient certainty for controlling the speed of a heat releasing chemical reaction. When the uncertainty is deemed to be sufficient, it can be reduced, say by employing a more accurate thermostat. Or a probabilistic model can be designed to incorporate the uncertainty, say by controlling the average speed of the reaction. 1 Example: Outcomes of a dice rolling experiment are 1, 2, 3, 4, 5 and 6. For a fair dice, each outcome is (sufficiently) uncertain. Each outcome of an experiment can be denoted generically by w or indexed as wi for specificity. Often these outcomes are minimal outcomes that cannot be or are not preferred to be separated into several other outcomes. Such minimal outcomes can be called elementary outcomes. Two elementary outcomes cannot occur at once, so elementary outcomes are mutually exclusive. Then elementary outcomes can be collected to obtain a set of outcomes W, that is generally called the sample space. Example: For the experiment of updating the accounting principles, w1 = K, w2 = I, w3 = D and W = fKg [ fIg [ fDg = fK, I, Dg. Example: For the experiment of rolling a dice, wi = i for i = 1, . , 6 and W = f1, 2, 3, 4, 5, 6g. An event is a collection of the outcomes of an experiment. So, an event is a subset of the sample space, i.e., a non-empty event A has w 2 A ⊆ W for some w. An event can be empty and is denoted by A = Æ. We are not interested in impossible events in practice, the consideration of Æ is useful for theoretical construction of probability models. Example: For the experiment of updating the accounting principles, the event of not increasing the reporting requirements can be denoted by fK, Dg. Example: For the experiment of rolling a dice, the event of an even outcome is f2, 4, 6g and the event of no outcome is Æ. Example: Consider the collision of two hydrogen atoms on a plane. One of the atoms is stationary at the origin and is hit by another moving from the left to right. After the collision, the atom moving from left to right can move to the 1st, 2nd, 3rd or 4th quadrant. The sample space for the movement of this atom is f1, 2, 3, 4g and the event that it bounces back (moves from right to left after the collision) is f2, 3g. Since an event corresponds to a set in the sample space, we can apply set operations on events. In particular, for two events A and B, we can speak of intersection, union, set difference. If the intersection of two events is empty, they are called disjoint: If A \ B = Æ, then A and B are disjoint. Example: In an experiment of rolling a dice twice, we can consider the sum and the multiplication of the numbers in the first and second rolls. Let A be the event that the multiplication is odd; B be the event that the sum is odd; C be the event that both multiplication and sum are odd; D be the event that both multiplication and sum are even. The outcomes in A have both the first and second number odd, while the outcomes in B have an odd number and an even number. Hence no outcome can be in both event A and B, which turn out to be disjoint events: A \ B = Æ = C. To be in D, an outcome must have both numbers even. In each outcome, either both numbers are odd (so multiplication is odd and sum is even =) A); or one is odd while the other is even (so multiplication is even and sum is odd =) B); or both numbers are even (so multiplication and sum are even =) D). Hence, A [ B [ D = W. To convince yourself further, you can enumerate each outcome and see whether it falls in A or B or D by completing Table 1. Some experiments are truly physical but their outcome is treated as a random variable in probability contexts. A good examples are rolling a die and flipping a coin. If we put a coin head up on our thumb and throw it up with no spin, it comes down head up. Then there is no flip or randomness. Some bakery cooks flatten the dough bread an throw it in the air without any flips1. Similarly, good players can consistently 1Indian cook throws dough for six metres https://www.youtube.com/watch?v=VhoiBIdz dU 2 Table 1: Sample space W for two dice rolls is composed of pairs (i, j) for i, j = 1 . 6. What is shown in each cell below are (multiplication=ij,sum=i + j) and the associated event. Second Roll 1 2 3 4 5 6 1 (1, 2), A (2, 3), B (3, 4), A (4, 5), B (5, 6), A (6, 7), B 2 (2, 3), B (4, 4), D (6, 5), B (8, 6), D (10, 7), B (12, 8), D First 3 (3, 4), (6, 5), (9, 6), (12, 7), (15, 8), (18, 9), Roll 4 (4, 5), (8, 6), (12, 7), (16, 8), (20, 9), (24, 10), 5 6 throw a frisbee or a football without a flip to facilitate a catch. We readily accept the absence of random- ness or a flip in the throws of flattened doug, friebee and football but make numerous examples based on randomness in throws of a coin. Perhaps, there is less rasndomness in coin flips than we are used to. Example: Suppose a coin is flipped with a vertical speed of vj and stays in the air for tair = 2vj/g seconds where g is the gravitational acceleration constant. The same coin has the angular speed of v◦ given in terms of revolutions per second. Then the coin makes v◦tair revolutions in the air. If the coin has head up and is thrown up to make 1 revolution in the air, it comes down as head up. If it makes (1/4, 3/4) revolutions in the air, it comes down as tail. In general, it comes down tail after (k + 1/4, k + 3/4) revolutions for k = 0, 1, 2 . Said differently, v◦tair 2 (k + 1/4, k + 3/4) for k = 0, 1, 2 . guarantees an outcome of tail. To start with head up and end with tail up (H ! T), we need to relate vertical and angular speeds to each other: vj 1 1 3 H ! T if v◦ 2 k + , k + for k = 0, 1, 2 . g 2 4 4 Figure 1 shows the regions of speed that maintain head (H ! H) or switch head to tail (H ! T). These regions are separated by hyperbolas of the form v◦vj = constant. From our experience, we can say that a coin stays in the air for about half a second, so vj/g ≈ 0.25 and the left-hand side of Figure 1 is more relevant in practical applications than its right-hand side.
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