Are service contracts for electronics just a scam?

Should your lottery ticket contain numbers greater than 31?

How do you know when he or she is “the one”?

How Math Can Save Your Life shows you how to use basic arithmetic to answer these and many other questions that come up in everyday life. You’ll discover how simple math can make you lots of money, keep you safe, and even save the world. Not bad for something you learned back in grade school.

Filled with practical, indispensable guidance you can put to work every day, this book will safeguard your wallet and enrich every aspect of your life.

“Even if you hated math in school, you’ll like this book. Jim Stein presents math the way I wish my teachers had: as a practical tool that can be used to solve everyday problems in the real world. Using down-to-earth langauge and real-life examples, Stein shows how even quick, back-of-the-envelope math can help us avoid costly errors.”
—Joseph T. Hallinan, author of Why We Make Mistakes

“Stein pulls off a literary hat trick by writing a book about mathematics that is fun, friendly and factual. It’s the definitive answer to the student’s question, ‘When will I ever need this stuff?'”
—Leonard Wapner, author of The Pea and the Sun: A Mathematical Paradox

Learn how a little math can help:

• Avert disasters

  • Beat the bookies

    • Boost your grades

      • Extend your life expectancy

        • Fix the economy

          • Improve your love life

            • Make you rich

              • Win arguments

                • And much more!

James D. Stein is the author of How Math Explains the World: A Guide to the Power of Numbers, From Car Repair to Modern Physics and is a professor of mathematics at California State University, Long Beach. A graduate of Yale University and the University of California, Berkeley, he has taught college math for more than forty years.

How Math Can Save Your Life

(And Make You Rich, Help You Find The One, and Avert Catastrophes)By James D. Stein

John Wiley & Sons

Copyright © 2010 John Wiley & Sons, Ltd
All right reserved.
ISBN: 978-0-470-43775-9

Chapter One

The Most Valuable Chapter You Will Ever Read

Are service contracts for electronics and appliances just a scam?

* * *

How likely are you to win at roulette?

* * * Is it worth going to college?

What constitutes value? On a philosophical level, I’m not sure; what’s valuable for one person may not be for others. The most philosophically valuable thing I’ve ever learned is that bad times are always followed by good times and vice versa, but that may simply be a lesson specific to yours truly. On the other hand, if this lesson helps you, that’s value added to this chapter. And if this chapter helps you financially, even better-because there is one universal common denominator of value that everyone accepts: money.

That’s why this chapter is valuable, because I’m going to discuss a few basic concepts that will be worth tens of thousands-maybe even hundreds of thousands-of dollars to you. So let’s get started.

Service Contracts: This Is Worth Thousands of Dollars

A penny saved is still a penny earned, but nowadays you can’t even slip a penny into a parking meter-so let me make this book a worthwhile investment by saving you a few thousand dollars. The next time you go to buy an appliance and the salesperson offers you a service contract, don’t even consider purchasing it. A simple table and a little sixth-grade math should convince you.

Suppose you are interested in buying a refrigerator. A basic model costs in the vicinity of $400, and you’ll be offered the opportunity to buy a service contract for around $100. If anything happens to the refrigerator during the first three years, the store will send a repairman to your apartment to fix it. The salesperson will try to convince you that it’s cheap insurance in case anything goes wrong, but it’s not. Let’s figure out why. Here is a table of how frequently various appliances need to be repaired. I found this table by typing “refrigerator repair rates” into a search engine; it’s the 2006 product reliability survey from Consumer Reports National Research Center. It’s very easy to read: the top line tells you that 43 percent of laptop computers need to be repaired in the first three years after they are purchased.

Use this chart, do some sixth-grade arithmetic, and you can save thousands of dollars during the course of a lifetime. For instance, with the refrigerator service contract, a refrigerator with a top-and-bottom freezer and no icemaker needs to be repaired in the first three years approximately 12 percent of the time; that’s about one time in eight. So if you were to buy eight refrigerators and eight service contracts, the cost of the service contracts would be 8 $100 = $800. Yet you’d need to make only a single repair call, on average, which would cost you $200. So, if you had to buy eight refrigerators, you’d save $800 – $200 = $600 by not buying the service contracts: an average saving of $600/8 = $75 per refrigerator. Admittedly, you’re not going to buy eight refrigerators-at least, not all at once. Even if you buy fewer than eight refrigerators over the course of a lifetime, you’ll probably buy a hundred or so items listed in the table. Play the averages, and just like the casinos in Las Vegas, you’ll show a big profit in the long run.

You can save a considerable amount of money by using the chart. There are basically two ways to do it. The first is to do the computation as I did above, estimating the cost of a service call (I always figure $200-that’s $100 to get the repairman to show up and $100 for parts). The other is a highly conservative approach, in which you figure that if something goes wrong, you’ve bought a lemon, and you’ll have to replace the appliance. If the cost of the service contract is more than the average replacement cost, purchasing a service contract is a sucker play.

For instance, suppose you buy a microwave oven for $300. The chart says this appliance breaks down 17 percent of the time-one in six. To compute the average replacement cost, simply multiply $300 by 17/100 (or 1/6 for simplicity)-the answer is about $50. If the service contract costs $50 or more, they’re ripping you off big-time. Incidentally, note that a side-by-side refrigerator with icemaker and dispenser will break down three times as often as the basic model. How can you buy something that breaks down 37 percent of the time in a three-year period? I’d save myself the aggravation and do things the old-fashioned way, by pouring water into ice trays.

Finally, notice that TVs almost never break down. I had a 25-inch model I bought in the mid-eighties that lasted seventeen years. Admittedly, I did have to replace the picture tube once. Digital cameras are pretty reliable, too.

The long-term average resulting from a course of action is called the expected value of that action. In my opinion, expected value is the single most bottom-line useful idea in mathematics, and I intend to devote a lot of time to exploring what you can do with it. In deciding whether to purchase the refrigerator service contract, we looked at the expected value of two actions. The first, buying the contract, has an expected value of $100; the minus sign occurs because it is natural to think of expected value in terms of how it affects your bottom line, and in this case your bottom line shows a loss of $100. The second, passing it up, has an expected value of $25; remember, if you bought eight refrigerators, only one would need a repair costing $200, and $200/8 = $25. In many situations, we are confronted with a choice between alternatives that can be resolved by an expected-value calculation. Over the course of a lifetime, such calculations are worth a minimum of tens of thousands of dollars to you-and, as you’ll see, they can be worth hundreds of thousands of dollars, or more, to you. This type of cost-effective mathematical projection can be worth millions of dollars to small organizations and billions to large ones, such as nations. It can even be used in preventing catastrophes that threaten all of humanity. That’s why this type of math is valuable.

Averages: The Most Important Concept in Mathematics

Now you know my opinion, but I’m not the only math teacher who believes this: averages play a significant role in all of the basic mathematical subjects and in many of the advanced ones. You just saw a simple example of an average regarding service contracts. Averages play a significant role in our everyday use of and exposure to mathematics. Simply scanning through a few sections of today’s paper, I found references to the average household income, the average per-screen revenue of current motion pictures, the scoring averages of various basketball players, the average age of individuals when they first became president, and on and on.

So, what is an average? When one has a collection of numbers, such as the income of each household in America, one simply adds up all of those numbers and divides by the number of numbers. In short, an average is the sum of all of the data divided by the number of pieces of data.

Why are averages so important? Because they convey a lot of information about the past (what the average is), and because they are a good indicator of the future. This leads us to the law of averages.

The Law of Averages

The law of averages is not really a law but is more of a reasonably substantiated belief that future averages will be roughly the same as past averages. The law of averages sometimes leads people to arrive at erroneous conclusions, such as the well-known fallacy that if a coin has come up heads on ten consecutive flips, it is more likely to come up tails on the next flip in order to “get back to the average.” There are actually two possibilities here: the coin is a fair coin that really does come up tails as often as it does heads (in the long run), in which case the coin is just as likely to come up heads as tails on the next flip; or the flips are somehow rigged and the coin comes up heads much more often than tails. If somebody asks me which way a coin will land that has come up heads ten consecutive times, I’ll bet on heads the next time-for all I know, it’s a two-headed coin.

Risk-Reward Ratios and Playing the Percentages

The phrases risk-reward ratio and playing the percentages are so much a part of the common vocabulary that we have a good intuitive idea of what they mean. The risk-reward ratio is an estimate of the size of the gain compared with the size of the loss, and playing the percentages means to select the alternative that has the most likely chance of occurring.

In common usage, however, these phrases are used qualitatively, rather than quantitatively. Flu shots are advised for the elderly because the risk associated with getting the flu is great compared with the reward of not getting it; that is, the risk-reward ratio of not getting a flu shot is high, even though we may not be able to see exactly how to quantify it. Similarly, on third down and seven, a football team will usually pass the ball because it is the percentage play: a pass is more likely than a run to pick up seven yards. There are two types of percentages: those that arise from mathematical models, such as flipping a fair coin, and those that arise from the compilation of data, such as the percentage of times a pass succeeds on third down and seven. When we flip a fair coin, we need not assume that in the long run, half of the flips will land heads and the other half tails, because that’s what is meant by “a fair coin.” If, however, we find out that 60 percent of the time, a pass succeeds on third down and seven, we will assume that in the long run this will continue to be the case, because we have no reason to believe otherwise unless the structure of football undergoes a radical change.

How, and When, to Compute Expected Value

The utility of the concept of expected value is that it incorporates both risk-reward ratios and playing the percentages in a simple calculation that gives an excellent quantitative estimate of the long-term average payoff from a given decision. Expected value is used to compute the long-term average result of an event that has different possible outcomes. The casinos of the world are erected on a foundation of expected value, and roulette wheels provide an easy way to compute an example of expected value. A roulette wheel has 36 numbers (1 through 36), half of which are red and half of which are black. In the United States, the wheel also has 0 and 00, which are green. If you bet $10 on red and a red number comes up, you win $10; otherwise, you lose your $10. To compute the expected value of your bet, suppose you spin the wheel so that the numbers come up in accordance with the laws of chance. One way to do this is to spin the wheel 38 times; each of the 38 numbers-1 through 36, 0, and 00-will come up once (that’s what I mean by having the numbers come up in accordance with the laws of chance). Red numbers account for 18 of the 38, so when these come up, you will win $10, a total of 18 $10 = $180. You will lose the other 20 bets, a total of 20 $10 = $200. That means that you lose $20 in 38 spins of the wheel, an average loss of a little more than $.52. Your expected value from each spin of the wheel is thus – $.52, and the casinos and all of those neon lights are built on your contribution and those of your fellow gamblers.

Expected value is frequently expressed as a percentage. In the preceding example, you have an average loss of about $.52 on a wager of $10. Because $.52 is 5.2% of $10, we sometimes describe a bet on red as having an expected value of – 5.2%. This enables us to compute the expected loss for bets of any size. Casinos know what the expected value of a bet on red is, and they can review their videotapes to see whether the actual expected value approximates the computed expected value. If this is not the case, maybe the wheel needs rebalancing, or some sort of skullduggery is taking place.

Expected value can be used only in situations where the probabilities and associated rewards can be quantified with some accuracy, but there are a lot of these. Many of the errands I perform require me to drive some distance; that’s one of the drawbacks of living in Los Angeles. Often, I have two ways to get there: freeways or surface streets. Freeways are faster most of the time, but every so often there’s an event (an accident or a car chase) that causes lengthy delays. Surface streets are slower, but one almost never encounters an event that turns a surface street into a parking lot, as can happen on the freeways. Nonetheless, like most Angelenos, I have made an expected-value calculation: given a choice, I take the freeway because on average I save time by doing so. It is not always necessary to perform expected-value calculations; simple observation and experience give you a good estimate of what’s happening, which is why most Angelenos take the freeway. You don’t have to perform the calculation for the roulette wheel, either; just go to Vegas, make a bunch of bets, and watch your bankroll dwindle over the long run.

Insurance: This Is Worth Tens of Thousands of Dollars

There’s a lot of money in the gaming industry, but it pales in comparison with another trillion-dollar industry that is also built on expected value. I’m talking about the insurance industry, which makes its profits in approximately the same way as the gaming industry. Every time you buy an insurance policy, you are placing a bet that you “win” if something happens that enables you to collect insurance, and that you “lose” if no such event occurs. The insurance company has computed the average value of paying off on such an event (think of a car accident) and makes certain that it charges you a large enough premium that it will show a profit, which will make your expected value a negative one.

Nonetheless, this is a game that you simply have to play. If you are a driver, you are required to carry insurance, and there are all sorts of insurance policies (life, health, home) that it is advisable to purchase, even though your expected value is negative-because you simply cannot afford the cost of a disaster. Despite that, there is a correct way to play the insurance game, and doing this is generally worth tens of thousands of dollars (maybe more) over the course of a lifetime.

Let’s consider what happens when you buy an auto insurance policy, which many people do every six months. My insurance company offers me a choice of a $100 deductible policy for $300 or a $500 deductible policy for $220. If I buy the $100 deductible policy and I get into an accident, I get two estimates for the repair bill and go to the mechanic who gives the cheaper estimate (this is standard operating procedure for insurance companies). The insurance company sends me a check for the amount of the repair less $100. If I had bought the $500 deductible policy, the company would have sent me a check for the amount of the repair less $500. It’s cheaper to buy the $500 deductible policy than the $100 deductible policy, because if I get in an accident, the insurance company will send me $400 less than I would receive if I’d bought the $100 deductible policy.

(Continues…)

Excerpted from How Math Can Save Your Lifeby James D. Stein Copyright © 2010 by John Wiley & Sons, Ltd. Excerpted by permission.
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