Probabilities, Odds, and Expectations

Transcription

1 MATH 110 Week 8 Chapter 16 Worksheet NAME Probabilities, Odds, and Expectations By the time we are finished with this chapter we will be able to understand how risk, rewards and probabilities are combined in a precise mathematical way that allows us to answer major questions like Should I really spend money buying lottery tickets? In order to do this, we will need to cover a lot of ground cover quickly Sample Spaces and Events In broad terms, probability is the quanification of uncertainty. To understand what this means, we need to start by formalizing the notion of uncertainty. There are two basic terms that we will need in order to begin this investigation: random experiment: sample space: Examples 1. One simple example of a random experiment is to toss a coin in the air once and see whether it lands heads or tails. What is the sample space? 2. Tossing a Coin Twice: Suppose we toss a coin twice and record the outcome of each toss in order. What is the sample space? 3. Suppose you roll a pair of dice simultaneously and record the outcome. What is the sample space? Notice that there are many different subsets of sample space. (Recall this from Section 2.4). 1

2 Events An event is a subset of the sample space and any subset of the sample space can be considered an event. Recall that in Section 2.4, we introduced the fact that a set with N elements has subsets The Multiplication Rule, Permutations, and Combinations In this section, we will introduce some basic mathematical tools that are used to determine the number of outcomes in a sample space (or an event) without having to list them all and then tally them one by one. We will use the word counting to describe the process of determining the number of elements in a set. Our methods of counting will be more sophisticated than those your learned in kindergarten. (In fact, complex methods of counting is still an active area of research known as combinatorics.) You are probably already familiar with the first rule of counting the multiplication rule: Example: Ice Cream Sabastian Joe s offers 3 different choices of cone and 16 different ice cream flavors. Using the multiplication rule, how many different choices for a single cone could you order? What if you had two different choices of toppings, how many different single cones with one topping could you order? How many different choices for a double cones could you order? What if you wanted to try two distinct flavors of ice cream, how many choices of double cones could you order? Before we can answer these questions we need to distinguish between two very important concepts. Permutations and Combinations Many counting problems boil down to counting the number of ways in which we can select groups of objects from a larger group of objects. We will need to discuss more sophisticated counting methods. We will also need to distinguish between two very similar but distinct concepts. Generally, we can distinguish them as: permutations and combinations Now let s return to our Ice Cream example. How many different choices of double cones could you order? What if you wanted to try two distinct flavors of ice cream, how many choices of double cones could you order? Notice that in both of these examples, each time we were many a choice of one flavor from 16...and then another one flavor from 16. The question remains of how we can generalize this to a choice of r objects from a set of n objects. We need a more specific definition of these two concepts to address this question: 2

3 permutation: combination: Example: The Lottery A typical lottery drawing consists of taking a few balls out of a large set of numbered balls inside a spinning container called a hopper. The range of numbers on the ball and the number of balls drawn from the hopper can vary with various kinds of lotteries. As you well know, if the numbers drawn from the hopper match the numbers on your lottery ticket, you win the jackpot. Suppose that, in this lottery, first, five balls are drawn from a hopper containing white balls numbered 1 through 56. Then one ball is drawn from a different hopper containing gold balls numbered 1 through 46. How many possible outcomes are there? In our ice cream and lottery examples, the order of the scoops and ball choices did not matter. Let s consider an example where the order does matter. Example: A set of reference books consists of eight volumes numbered 1 through 8. How many ways can the eight books be arranged on a shelf? 16.3 Probabilities and Odds Referring back to our coin toss, we already know what the probability of getting a heads is. The argument as to exactly how to interpret the statement the probability of X is such and such goes back to the late 1600s, it wasn t until the 1930s that a formal theory for dealing with probabilities was developed by Russian mathematician A. N. Kolmogorov. This theory has made probability one of the most useful and important concepts in modern mathematics. Example: Tossing a Coin Twice What is the probability of getting at least one heads when tossing two coins? What is the probability of tossing exactly one heads when tossing two coins? Probability Assignments What is a probability assignment? 3

4 There are two things to note about probabilities: Once a specific probability assignment is made on a sample space, the combination of the sample space and the probability assigment is called the probability space. The following summarizes the key elements of a probability space: sample space: probability assignment: Equiprobable Space One of the most common uses of randomness in the real world is as a mechanism to guarantee fairness. What is an equiprobable space? In equiprobable spaces, calculating probabilities of the events becomes simply a matter of counting. probability of an event: Example: Rolling a Pair of Dice Imagine you are playing a game that involves rolling a pair of dice and the only thing that matters is the total of the two numbers rolled. What is the sample space in this situation? Determine the probability of each of these events. What is the probability of at least one of the dice coming up and ace (as a 1 )? 4

5 This example introduces a few important concepts: complementary events: independent events: multiplication principle for independent events: Example: 1. Imagine you are are playing a game in which you roll an honest pair of die four times. If at least one come up an ace, you are a winner. What is the probability that you will win? item Now imagine you are in a game where you roll an honest die four times. If at least one of your rolls comes up an ace, you are a winner. What is the probability that you will win? Odds Probabilities and odds are not identical concepts but they are related. What are odds? Example: Rolling a Pair of Dice Suppose you are playing a game in which you roll a pair of dice. In this game, when you roll a natural (i.e. roll a 7 or an 11) you automatically win. What is the probability of rolling a natural? What are the odds of rolling a natural? 16.4 Expectations Weighted Averages What is a weighted average? 5

6 Example: Computing Class Scores Imgine you are a student in a Math 101 class. The grading for this class is based on two midterms, homework, and a final exam. You need a 90% or above to get an A in the course, use the following information to determine whether you will get one: Midterm 1 Midterm 2 Homework Final Exam Weight 20% 20% 25% 35% Possible Points Your Scores What is an expectation? Example: Guessing Answers in the SAT In the multiple choice section of the SAT each question has five possible answers (A, B, C, D, and E). A correct answer is worth 1 point, and to discourage indiscriminate guessing, an incorrect answer carries a penalty of 1/4 point. Imagine you are taking the SAT and are facing a multiple choice question for which you have no clue as to which of the five choices might be the right answer. You can either play it safe andleave it blank or take a wild guess. In the latter case, the upside of getting 1 point must be weighed against the downside of getting -1/4 point. What should you do? Now assume that you can safely rule out one of the choices, say choice E. In this case, what is the expected payoff? 16.5 Measuring Risk In many real-life situations, we face decisions that can have many different potential consequences some good, some bad, some neutral. These kind of decisions are often quite hard to make because there are so many intangibles, but sometimes the decision comes down to a simple question: Is the reward worth the risk? 6

7 Example: Raffles and Fund-Raisers At a fund-raiser, the raffle tickets are going for $2. In this raffle, the organizers will draw one grand-prize winner worth $500, four second-prize winners worth $50 each and fifteen third-prize winners worth $20 each. Sounds like a pretty good deal for a $2 investment, but is it? Suppose 2000 raffle tickets are sold and lets assume a raffle ticket can only win one of the prize, what is the expected gain? A game is considered a fair game when no player has a built-in advantage over another player in the game. In the case of a game between a player and the house a fair game is one where expected gain is $0. Let s return to the lottery example we discussed earlier. Read and discuss Example: Mega Millions on p.503. Homework: p.508 # 2, 4, 8, 18, 20, 28, 30, 34, 42, 58, 62, 66, 68, 74, 81, 86, 88 7