A Field Guide to Lies: Critical Thinking in the Information Age by Daniel J. Levitin
Authors: Daniel J. Levitin
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testimony, but not bad. (If you’re wondering why they don’t just ask you a bunch of short-answer, fill-in questions, where you have to provide the entire answer yourself, instead of using multiple choice, it’s because there are too many variants of correct answers. Do you refer to your credit card as being with Chase, Chase Bank, or JPMorgan Chase? Did you live on North Sycamore Street, N. Sycamore Street, or N. Sycamore St.? You get the idea.)
When the Probability of Events Is Informed by Other Events
The multiplication rule only applies if the events are independent of one another. What events are not independent? The weather, for example. The probability of it freezing tonight and freezingtomorrow night are not independent events—weather patterns tend to remain for more than one day, and although freak freezes are known to occur, your best bet about tomorrow’s overnight temperatures is to look at today’s. You could calculate the number of nights in the year in which temperatures drop below freezing—let’s say it’s thirty-six where you live—and then state that the probability of a freeze tonight is 36 out of 365, or roughly 10 percent, but that doesn’t take the dependencies into account. If you say that the probability of it freezing two nights in a row during winter is 10% × 10% = 1% (following the multiplication rule), you’d be underestimating the probability because the two nights’ events are not independent; tomorrow’s weather forecast is informed by today’s.
The probability of an event can also be informed by the particular sample that you’re looking at. The probability of it freezing tonight is obviously affected by the area of the world you’re talking about. That probability is higher at the forty-fourth parallel than the tenth. The probability of finding someone over six foot six is greater if you’re looking at a basketball practice than at a tavern frequented by jockeys. The subgroup of people or things you’re looking at is relevant to your probability estimate.
Conditional Probabilities
Often when looking at statistical claims, we’re led astray by examining an entire group of random people when we really should be looking at a subgroup. What is the probability that you have pneumonia? Not very high. But if we know more about you and your particular case, the probability may be higher or lower. This is known as a conditional probability .
We frame two different questions:
What is the probability that a person drawn at random from the population has pneumonia?
What is the probability that a person not drawn at random, but one who is exhibiting three symptoms (fever, muscle pain, chest congestion) has pneumonia?
The second question involves a conditional probability. It’s called that because we’re not looking at every possible condition, only those people who match the condition specified. Without running through the numbers, we can guess that the probability of pneumonia is greater in the second case. Of course, we can frame the question so that the probability of having pneumonia is lower than for a person drawn at random:
What is the probability that a person not drawn at random, but one who has just tested negative for pneumonia three times in a row, and who has an especially robust immune system, and has just minutes ago finished first place in the New York City Marathon, has pneumonia?
Along the same lines, the probability of you developing lung cancer is not independent of your family history. The probability of a waiter bringing ketchup to your table is not independent of what you ordered. You can calculate the probability of any person selected at random developing lung cancer in the next ten years, or the probability of a waiter bringing ketchup to a table calculated over alltables. But we’re in the lucky position of knowing that these events are dependent on other behaviors. This allows us to narrow the population we’re studying in order to obtain a
When the Probability of Events Is Informed by Other Events
The multiplication rule only applies if the events are independent of one another. What events are not independent? The weather, for example. The probability of it freezing tonight and freezingtomorrow night are not independent events—weather patterns tend to remain for more than one day, and although freak freezes are known to occur, your best bet about tomorrow’s overnight temperatures is to look at today’s. You could calculate the number of nights in the year in which temperatures drop below freezing—let’s say it’s thirty-six where you live—and then state that the probability of a freeze tonight is 36 out of 365, or roughly 10 percent, but that doesn’t take the dependencies into account. If you say that the probability of it freezing two nights in a row during winter is 10% × 10% = 1% (following the multiplication rule), you’d be underestimating the probability because the two nights’ events are not independent; tomorrow’s weather forecast is informed by today’s.
The probability of an event can also be informed by the particular sample that you’re looking at. The probability of it freezing tonight is obviously affected by the area of the world you’re talking about. That probability is higher at the forty-fourth parallel than the tenth. The probability of finding someone over six foot six is greater if you’re looking at a basketball practice than at a tavern frequented by jockeys. The subgroup of people or things you’re looking at is relevant to your probability estimate.
Conditional Probabilities
Often when looking at statistical claims, we’re led astray by examining an entire group of random people when we really should be looking at a subgroup. What is the probability that you have pneumonia? Not very high. But if we know more about you and your particular case, the probability may be higher or lower. This is known as a conditional probability .
We frame two different questions:
What is the probability that a person drawn at random from the population has pneumonia?
What is the probability that a person not drawn at random, but one who is exhibiting three symptoms (fever, muscle pain, chest congestion) has pneumonia?
The second question involves a conditional probability. It’s called that because we’re not looking at every possible condition, only those people who match the condition specified. Without running through the numbers, we can guess that the probability of pneumonia is greater in the second case. Of course, we can frame the question so that the probability of having pneumonia is lower than for a person drawn at random:
What is the probability that a person not drawn at random, but one who has just tested negative for pneumonia three times in a row, and who has an especially robust immune system, and has just minutes ago finished first place in the New York City Marathon, has pneumonia?
Along the same lines, the probability of you developing lung cancer is not independent of your family history. The probability of a waiter bringing ketchup to your table is not independent of what you ordered. You can calculate the probability of any person selected at random developing lung cancer in the next ten years, or the probability of a waiter bringing ketchup to a table calculated over alltables. But we’re in the lucky position of knowing that these events are dependent on other behaviors. This allows us to narrow the population we’re studying in order to obtain a
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