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UW-Madison STAT 371 - Chapter 1 Probability

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0Chapter 1Probability1.1 Getting StartedThese notes will explore how the discipline of Statistics helps scientists learn about the world.There are two main areas in which Statistics helps:• Validity. Proper use of Statistics helps a scientist learn things that are true; improper use canlead to a scientist learning things that are false.• Efficiency. Proper use of Statistics can help a scientist learn faster, or with less effort or at alower cost.The first tool you need to become a good statistician is a cursory understanding of probability.I begin with one of my favorite quotes from a favorite source.Predictions are tough, especially about the future.—Yogi Berra.Probability theory is used by mathematicians, scientists and statisticians to quantify uncertaintyabout the future.We begin with the notion of a chance mechanism. This is a two-word technical expression. Itis very important that we use technical expressions exactly as they are defined. In every day lifeyou may have several meanings for some of your favorite words, for example phat, but in this classtechnical expressions have a unique meaning. In these notes the first occurrence of a technicalexpression/term will be in bold-faced type.Both words in ‘chance mechanism’ (CM) are meaningful. The second word reminds us thatthe CM, when operated, produces an outcome. The first word reminds us that the outcome cannotbe predicted with certainty.Several examples will help.1. CM: A coin is tossed. Outcome: The face that lands up, either heads or tails.2. CM: A (six-sided) die is cast. Outcome: The face that lands up, either 1, 2, 3, 4, 5 or 6.13. CM: A man with AB blood and a woman with AB blood have a child. Outcome: The bloodtype of the child, either A, B or AB.4. CM: The next NFL Super Bowl game. Outcome: The winner of the game, which could beany one of the 32 NFL teams.The next idea is the sample space, usually denoted by S. The sample space is the collectionof all possible outcomes of the CM. Below are the sample spaces for each CM listed above.1. CM: Coin. S = {H, T }.2. CM: Die. S = {1, 2, 3, 4, 5, 6}.3. CM: Blood. S = {A, B, AB}.4. CM: Super Bowl. S = A list of the 32 NFL teams.An event is a collection of outcomes; that is, it is a subset of the sample space. Events aretypically denoted by upper case letters, usually from the beginning of the alphabet. Below aresome events for each CM listed above.1. CM: Coin. A = {H}, B = {T }.2. CM: Die. A = { 5, 6}, B = {1, 3, 5}.3. CM: Blood. C = {A, B}.4. CM: Super Bowl. A = {Vikings, Packers, Bears, Lions}.Sometimes it is convenient to describe an event with words. As examples of this: For the die,event A can described as ‘the outcome is larger than 4,’ and event B can be described as ‘theoutcome is an odd integer.’ For the Super Bowl, event A can described as ‘the winner is from theNFC North Division.’Here is where I am going with this: Before a CM is operated, nobody knows what the outcomewill be. In particular, for any event A that is not the entire sample space, we don’t know whetherthe outcome will be a member of A. After the CM is operated we can determine/see whether theactual outcome is a member of an event A; if it is, we say that the event A has occurred; if not, wesay that the event A has not occurred. Below are some examples for our CM’s above.1. CM: Coin. If the coin lands heads, then event A has occurred and event B has not occurred.2. CM: Die. If the die lands 5, both A and B have occurred. If the die lands 1 or 3, B hasoccurred, but A has not. If the die lands 6, A has occurred, but B has not. Finally, if the dielands 2 or 4, both A and B have not occurred.3. CM: Blood. If the child has AB blood, then the even C has not occurred.4. CM: Super Bowl. If the Packers win the Super Bowl, then the event A has occurred.2Before the CM is operated, the probability of the event A, denoted by P (A), is a number thatmeasures the likelihood that A will occur. This incredibly vague statement raises three questionsthat we will answer.1. How are probabilities assigned to events?2. What are the rules that these assignments must obey?3. If I say, for example, that P (A) = 0.25, what does this mean?First, the assignment of probabilities to events always is based on assumptions about the op-eration of the world. As such, it is a scientific, not a mathematical exercise. There are alwaysassumptions, whether they are expressed or tacit; implicit or explicit. My advice is to always doyour best to be aware of any assumptions you make. (This is, I believe, good advice for outsidethe classroom too.)The most popular assumption for a CM is the assumption of the equally likely case (ELC).As the name suggests, in the ELC we assume that each possible outcome is equally likely to occur.Another way to say this is that it is impossible to find two outcomes such that one outcome ismore likely to occur than the other. I will discuss the ELC for each CM we have considered in thissection.1. CM: Coin. If I select an ordinary coin from my pocket and plan to toss it, I would assumethat the two outcomes, heads and tails, are equally likely to occur. This seems to be a popu-lar assumption in our culture because tossing a coin is often used as a way to decide whichof two persons/teams is allowed to make a choice. For example, football games typicallybegin with a coin toss and the winner gets to make a choice involving direction of attack orinitial possession of the ball. Note, however, that I would not make this assumption withoutthinking about it. In particular, the path of a coin is governed by the laws of physics and pre-sumably if I could always apply exactly the same forces to the coin it would always land thesame way. I am an extremely minor acquaintance of a famous person named Persi Diaconis.Persi has been a tenured professor at Stanford, Harvard, Cornell and Stanford again, and hewas a recipient of a MacArthur Foundation no-strings-attached genius fellowship a numberof years ago. More relevant for this discussion is that while a teenager, Persi worked as asmall acts magician. Thus, it is no surprise to learn that Persi has unusually good control ofhis hands and reportedly can make heads much more likely than tails when he tosses a coin.My willingness to assume that heads and tails are equally likely when I toss a coin reflectsmy belief about how coins are balanced and my limited ability to control my hands.2. CM: Die. Again, if I take an ordinary die from a board game I am


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UW-Madison STAT 371 - Chapter 1 Probability

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