Fundamentals of Probability: Sample and Event Spaces
Understanding the basics of probability involves defining sample and event spaces, interpreting probability models, and applying these concepts to solve problems. By the end of the lecture, you will be able to identify sample and event spaces in probability questions and create meaningful probability models for analysis and decision-making.
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Learning Objectives By the end of this lecture, you should be able to: Define the term sample space and event space. Be able to identify those spaces through careful reading of a probability question. Create and interpret a probability model.
Yeah, yeah Often we look at topics that seem like common sense and say to ourselves, yeah, yeah, I get that . BE CAREFUL! Often there are many ways in which we think we understand something, but there still remain (many!!) gaps in our knowledge and understanding. This is not only true of statistics happens in all kinds of places/studies. However, it is a particularly common pitfall in stats. Be sure to review the concepts and complement them with lots of problems. There are no shortcuts if there were I promise that I would tell them to you! Review of the lectures, and lots of problems is the only way to learn this material.
Randomness and probability A phenomenon is random if individual outcomes are uncertain. And yet, if the probabilities are available, then given a large number of repetitions, you can expect to see a regular distribution of outcomes. For example, a single individual flip of a coin is random. However given a large number of coin-flips we can expect to see about 50% heads and 50% tails. The Concept: The expected outcome of a random phenomenon only applies to a very long series of repetitions. For example, the probability of flipping a heads is almost exactly 50% given a large number of repetitions (flips). (And assuming that the coin is fair / equally weighted on both sides).
Examples If you flip a coin, how often would you expect to get a heads? Answer: 50% of the time. If you flip a coin 10 times, can you safely expect to get 5 heads and 5 tails? Answer: No! In fact, try it The reason is that 10 flips of a coin is a relatively low number. You can only expect to get values close to 50% heads and 50% tails if you flip the coin over a large series of repetitions. If you roll a die, how often would you expect to get a 4? Answer: 1 in 6 times. Suppose you roll the die six times, can you safely assume that you will see a 4 one and only one time? Again of course not! You can only expect to see the 4 show up exactly 1/6 times over many rolls of the die.
Terms / Models Learn em!!! Sometimes we can intuitively figure things out. What is the probability of drawing an Ace of Spades from a deck of cards? Answer: 1/52 What is the probability of rolling a die and getting a 6 if on the previous roll you got a 1? Answer: 1/6 Sometimes, however, things become much more subtle and/or complicated. This is why it is important to become very familiar with terms and the concepts they refer to. In particular: probability model independent vs non-independent events disjoint vs non-disjoint events conditional probability Always focus on applying the proper concept/model and you will have a much better chance at understanding the concepts and coming up with the correct answers. Remember, it s the concept that matters. If you don t have the concept, you risk incorrect statistics.
Sample Space The sample space refers to the list of all possible outcomes of an event. Determining the sample space sometimes requires careful reading of the question. Key Point: Make sure that you correctly identify the sample space and have it clear in your mind. If you make a mistake in identifying your sample space, you will end up with an incorrect solution to the question. What is the sample space for the number of heads in three flips of the coin? Note that we are asking for the number of heads in three flips. S(# Heads) = {0, 1, 2, 3} What is the sample space for the number of 6s in two rolls of a die? Note that we are asking about how many 6s might occur in two rolls S(# 6s) = {0, 1, 2} What is the sample space for the number of SuperBowls that could be won by the Chicago Bears in the next 10 years? S(# SuperBowl Wins) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} What is the sample space for a randomly selected day of the week? S(Random Day of the Week) = {Sun, Mon, Tue, Wed, Thu, Fri, Sat} Note: The best way to make sure you have correctly identified the sample space, is to make sure that you read the question carefully!
Sample Space The phrasing of the question determines the sample space. H - HHH A. A basketball player shoots three free throws. What are the possible sequences of hits (H) and misses (M)? H M - HHM H S = { HHH, HHM, HMH, HMM, MHH, MHM, MMH, MMM } H - HMH M M - HMM M B. A basketball player shoots three free throws. What is the number of baskets made? S = { 0, 1, 2, 3 } C. A nutrition researcher feeds a new diet to a young male white rat. What are the possible outcomes of weight gain (in grams)? S = [0, [ (the [ means that infinity is excluded)
Probability of Outcomes in the Sample Space The sample space shows all possible outcomes. Each of these outcomes has some probability of occurring. We notate this outcome as P( .) Egg: Looking at a single coin flip. Let X = the result of the flip. What is P(X=heads)? Answer: P(X=Heads) = 0.5 Egg: Looking at two coin flips, Let X = the number of heads. What is P(X=2)? Note that this question asks about the probability of getting 2 heads on two coin flips. Answer is 0.25 we will explain how we got this number later. For the moment, an important point here, is to note the terminology/symbology being used. The various possible outcomes of a sample space can and often DO have different probabilities. Outcomes of 1 flip of a coin: S = {Heads, Tails}. In this case, each of the two possible events has the same probability of occurring. P(Heads) = 0.5, P(Tails) = 0.5 An at-bat baseball player s outcomes for getting a hit or getting an out has the sample space S = {hit, out}. Note that each outcome in this sample space has a different probability. For example: P(Hit) = 0.294, P(Out) = 0.706 A good professional basketball player may hit 70% of free throws. S = {hit, miss} P(Basket) = 0.7, P(Miss) = 0.3
Restated Different outcomes in a sample space can and often do have different probabilities of occurring What is sample space for getting hit by lightning in any one year? S = {Get Hit by Lightning, Do NOT get hit by lightning} What is the probability of each of these outcomes? Let X = the possible outcomes of the sample space P(X=get hit) = About 0.00014% P(X=do not get hit by lightning) = About 99.99986%
Outcomes / Events It is important to be able to CLEARLY identify the sample space and also the event(s) you are interested in. If you do not read the question with some care, you can easily mis-identify one or both of these. Without proper identification of the sample space and the event, you re gonna end up with the wrong answer! An event is a list of the particular outcomes that you are interested in. An event is typically some subset of the sample space. Notice that the event is NOT the same thing as the probability. The event is simply the result you are interested in calculating a probability for. The probability refers to the likelihood of that event occurring.
Example: Identify the Outcomes and Events REVIEW: The sample space is a list of all possible outcomes. An event is a list of any particular outcomes that you are interested in. For the following two examples, define the sample space and the event. Then rephrase the probability question using appropriate symbols: If I roll a single die what is the probability that I will roll a 1? Sample space = {1, 2, 3, 4, 5, 6} Event = {1} Let X = the outcome of the roll. What is P(X=1)? If I roll two die, what is the probability the sum will be a 7 or an 11? We are talking about the sum in two rolls of the die. So in this case: Sample space = {2,3,4,5,6,7,8,9,10,11,12} Event = {7, 11} Let X = the sum of the two die. What is P(X=7 or X=11)? Again: An event is always some subset of the sample space. Notice that the event is NOT the same thing as the probability. The event is simply the result you are interested in calculating a probability for. The probability refers to the likelihood of that event occurring.
Probability Models Probability models describe, mathematically, the outcome of random processes. A probability model consists of two parts: 1) Sample Space ( S ): This is the set, of all possible outcomes. 2) A probability for each of the possible outcomes in the sample space. Example: Probability Model of the Heads/Tails outcome in a single Coin Toss Sample Space: Heads Tails Probability: 0.5 0.5
Probability Model for the result of a single die roll Sample Space (All possible outcomes) 1 2 3 4 5 6 Probability (For each outcome) 1/6 1/6 1/6 1/6 1/6 1/6
Probability Model Example Example: A couple wants three children. What are the numbers of girls they could end up with? What are the probabilities for each outcome? Create a probability model. Also: Be sure to note and be comfortable with the way we use variables and symbols such as P(X=2) Sample space S = {0, 1, 2, 3} Let X = the number of girls. P(X = 0) = P(BBB) = 1/8 P(X = 1) = P(BBG or BGB or GBB) = P(BBG) + P(BGB) + P(GBB) = 3/8 P(X = 2) = P(BGG or GBG or GGB) = P(BBG) + P(BGB) + P(GBB) = 3/8 P(X = 3) = P(GGG) = 1/8 Probability Model:
Terminology Example: How many girls might the couple have? We will use the probability model from the previous example. It is vitally important to clearly define what X (or whichever variable you want to use it does not have to be X!) represents. In this case, X is a count of the number of girls that the couple will end up with out of their 3 children. So if your statistics prof gives you the situation in the previous slide and then writes P(X=2) what exactly are you being asked? Answer: What is the probability of the couple ending up with exactly two girls? If you were asked P(X>2)? Answer: What is the probability of the couple ending up with exactly 3 girls? What if you were asked P(X>=2)? Answer: What is the probability that the couple ending up with either 2 girls or 3 girls? You must be very clear with this concept. Note that on this slide, I do not spend the time working out the mathematical answer. This is intended as a reminder to you that the most important thing to do first is to clearly understand what the question is asking and to understand precisely what the variables represent.
Coin Toss Example: S = {Head, Tail} Probability of heads = 0.5 Probability of tails = 0.5 Probability rules 1) Probabilities range from 0 (no chance of the event) to 1 (the event will always happen). Probability of getting a Head = 0.5 We write this as: P(Head) = 0.5 P(neither Head nor Tail) = 0 P(getting either a Head or a Tail) = 1 For any event A, 0 P(A) 1 2) The sum of the probabilities of all the possible outcomes (the sample space) must be exactly 1. Coin toss: S = {Head, Tail} P(head) + P(tail) = 0.5 + 0.5 =1 P(sample space) = 1 P(sample space) = 1
Example: A basketball free-thrower who makes the basket 72% of the time. : S = {Hit, Miss} P(Hit) P(Miss) = 0.28 P(Hit + Miss) = 1.0 Recall, all probabilities in the sample space must add up to 1. = 0.72 Again, note that the sum of all probabilities must add up to 1. In this case, the two outcomes, Hit & Miss add up to 1.
