to be used.A: We make mental categories of things that seem to “go together.” Here, both cough drops and VapoRub have a menthol smell (at least to me), so given that one is supposed to eat cough drops, I figured anything with a similar smell is supposed to be eaten as well.4 Explain why people are more likely to carry their umbrellas when they hear there is a “20% chance of rain” than when there is an “80% chance it will be dry.” Both phrases contain the same information, so why is there a difference in how people respond to them?A: This is an example of the framing effect. By hearing the word “rain,” people think about getting wet. By hearing the word “dry,” people do not think about getting wet. Therefore, the image of being wet prompts people to carry their umbrellas.
5 LeBron James, one of the best players in professional basketball, has challenged me to a game of one-on-one basketball. By using information about the law of small numbers, explain how I could maximize the likelihood that I beat James at his sport.A: Even the best basketball players will sometimes miss a shot. Even the worst basketball players will sometimes make a shot. Therefore, to maximize my chances of winning, which on the surface seem nonexistent, I would want to play just one shot against LeBron James. This is an application of the law of small numbers. Perhaps he’ll miss his shot and I will make mine and, thus, win. The longer the game goes, the more likely I am to lose because he is the better basketball player.
6 Suppose I made you the following offer: You pay me $4, and I will flip a legitimate coin for which you call “heads” or “tails.” If you call the flip correctly, I’ll pay you $10. If you call it incorrectly, you get nothing. How many times, if any, would you play this game with me? Explain your reasoning.A: If you want to make a lot of money, you should play this game as often as possible. On average, you will call the flip correctly 50% of the time. When you do, you win $10. When you don’t, you lose $4. Suppose you played this game twice, winning once and losing once. To play twice costs $8 ($4 each time). If you win only once, you walk away with $10, so you made $2. The more you play this game, the more money you will make. However, we know from the law of small numbers that you want to play it more than once or twice because it is possible you could end up losing money with a small number of flips. But over the course of many flips, you will win money.
7 What is the difference between a population and a sample?A: The population is larger than the sample. The sample is used to draw conclusions about the population, which is the entire group you want to learn about in a research study.
8 Why is blood pressure a variable?A: A variable is a characteristic that differs among members of a population. People have different blood pressure levels, so therefore it is a variable (and one of great interest to medical researchers).
Misunderstanding Connections Between Events
In addition to not using information about probability, we as human beings also have a need to perceive order in the world. Think about being at a party, and you suddenly find yourself around people you do not know. You know almost nothing about them (other than they are at the same party you are), so what do you say to them? You have almost no idea where to begin to strike up a conversation. Maybe you look at their clothes for some hint of what they are like. If someone is wearing a University of Florida T-shirt, you might ask them if they are from the state of Florida and perhaps mention you visited there (assuming that you did). You look for something, anything, to break the uneasiness of that situation. Even for the most socially outgoing among us, it is an awkward situation. So, you start looking to make connections with these people.
Figure 1.2 Two Equally Likely Outcomes
Our need to make connections is powerful. The world is much less stressful when it is predictable. Therefore, our minds seek to make connections between events in the world. We will highlight two of these tendencies now. The first tendency concerns perceiving connections that in fact do not exist. The second of these tendencies is a result of the first and concerns the misinterpretation of future events based on prior events.
Illusory correlations
To start a discussion of the first of these two tendencies, take a look at these two poker hands of cards in Figure 1.2. Which one is a player more likely to be dealt? A hand of 10 through ace, all of the same suit, feels highly unlikely. However, in reality, the odds of getting that hand are no lower than the odds of getting the other, seemingly random, hand of cards. We are wired as human beings to detect patterns in the world, and so it is here. We feel as though the 10 through ace is a more unusual hand than the other hand of cards. Statistically, however, the odds of getting either one are the same. This is an example of an illusory correlation. By “illusory,” we mean “not real” or an “illusion.” By correlation, we mean an “association” or “connection” between two behaviors or events. Much as an optical illusion is seeing something that is not present, an illusory correlation is perceiving a relationship when no relationship exists (Fiedler, 2000). We want the world to be a predictable, orderly place. So our minds impose order and logic even when order and logic do not exist.
Illusory correlation: tendency to perceive a relationship when no relationship really exists.
Let’s discuss some additional examples of illusory correlations. In college, I had a roommate, Alex, who was not only a nice guy, but smart, too. Alex worked hard and never took his natural intellectual ability for granted. He did well in all of his classes, and he went on to become an immigration lawyer in South Florida. We were roommates for our entire college careers, and he is still a good friend to this day. I am sure this information does not impress you. But here is what might impress you. Take a guess as to why Alex did so well in college and beyond. His intelligence? His hard work? Those would be reasonable guesses. Ask Alex, however, and he will tell you something different. Let me explain. In our first year of college, we had our first round of tests about five weeks into the semester. We both had two or three tests that week. Of course, as would become the norm, Alex did well on these assessments. About five weeks later, we had our second round of tests and papers coming due. Because Alex had done so well on tests and papers earlier in the semester, I asked him for some study tips. Here is what he told me: He told me that I could not wear his pair of “lucky socks.” They were his lucky socks, and no one else could wear them. He had worn them every day during the week of our first round of tests and papers, and because he did so well, he was going to wear them again when tests and papers came due. Of course, I thought he was joking, but no, it quickly became clear that in his mind, the reason he did well on tests was because of his lucky pair of socks. By believing, albeit erroneously, that his pair of socks aided his performance on his tests, Alex was able to feel as though he had gained some control over his environment. The next time he had a test, he just needed to wear that same pair of socks again, and he would do well.4
Photo 1.9a and 1.9b Does a “lucky” charm really help us get good grades?
If you have ever played a sport or been involved in the performing arts, did you ever have some sort of pre-performance routine that you felt you had to follow? Just like my college roommate and his belief that wearing a certain pair of socks contributed to his academic success, some highly accomplished professional athletes have pre-performance routines that they follow. For instance, three-time Ironman champion Chrissie Wellington wrote the Rudyard Kipling poem “If” on her water bottles before each event. Similarly, professional baseball player Justin Verlander reportedly used to eat Taco Bell® (Yum! Brands Inc., Louisville, KY) the night before the games in which he was the starting pitcher. Is there really a relationship between writing
