13.5: Statistics and Probability

Even when we are dealing with statistically significant statistics, we critical thinkers have to be on our guard not to be bamboozled by statistics. Which would you prefer, a drink that is 96% fat-free or one that is 4% fat? Most of us would prefer the first one, but we aren’t thinking critically here, because there is no difference in the two.

"If something happens to only one in a million people per day, and the population of the United States is 250 million, then you expect 250 amazing coincidences every day," says statistician Ivars Peterson. Some of these "miracles" will get reported publicly, and we media consumers will be awed, when we probably should not be.

Exercise \(\PageIndex{1}\)

If you look at the speed people are driving when they get in auto accidents, you will find that a much higher percentage of accidents occur at speeds below 70 miles per hour than at speeds over 100 miles per hour. Therefore, to be safe you should try to drive over 100 miles per hour.

You do want to be safe, don’t you? Or maybe you prefer living on “the edge.” Or maybe you saw through my silly recommendation about driving over 100 miles per hour. Can you say what is wrong with the reasoning other than that it is silly?

Answer

About the speeding, very few people drive that fast, so naturally there are few accidents at that speed; but the chances of having an accident when driving at that speed are astronomical compared to driving within the speed limit.

Let's turn from statistics to probability.

"Doubt is not a pleasant condition; but certainty is an absurd one." -Voltaire

Probability involves putting a number on the chance of an event taking place. The custom is that probability numbers must be on a scale from zero to one, with zero meaning the event definitely will not occur and with one meaning it definitely will. Most probabilities we are interested in fall somewhere between these two extremes.

Consider a game involving dice. When we roll a fair die, there are six possible outcomes, all equally likely. Suppose we are interested in the probability of getting a 5. That means that exactly one of the six possible outcomes is a success, giving a probability of 1/6. The fundamental principle here is straightforward. The probability of a successful outcome is always the ratio:

Number of successful outcomes divided by the total number of possible outcomes,

so long as every outcome is equally likely. If the outcomes are not equally likely, the math gets complicated.

Gamblers who bet on the outcome of the role of a fair die sometimes make the mistake of thinking that, if after ten or twenty rolls, a five has come up less than 1/6 of the time, then a five is "due," meaning that on the next roll a five is more likely than 1/6. This mistake in reasoning is called the gambler's fallacy. A five has the same probability regardless of the history of the die.

But all this was on the assumption that the die was "fair." Let's relax that assumption. Suppose someone shows you a coin with a head and a tail on it. You watch him flip it ten times and all ten times it comes up heads. What is the probability that it will come up heads on the eleventh flip? Let’s consider what three people would say.

A person who commits the gambler’s fallacy would tell you, “Tails is more likely than heads, since things have to even out and tails is due to come up.”

A math student would tell you, “We can’t predict the future from the past; individual trials are uncorrelated. So, the odds are still even.”

A professional gambler would say, “There must be something wrong with the coin or the way it is being flipped. I wouldn’t bet with the guy flipping it. However, on an even bet I’d bet someone else who isn’t a friend of the guy doing the flipping that heads will come up again.”

The professional gambler is the most sensible of these three people.

Notice how we apply our background knowledge to estimates of probability. Suppose you know you are reaching blindly into a container of white and black balls. Let’s suppose you sample the container, replacing the ball after each sample. You do this five times and get 4 white balls and 1 black ball. Then if you were asked whether a white ball or a black ball is likely next time, you’d say a white ball. But if you had background knowledge that the container has one white ball and 99 black balls, then you’d guess a black ball next time, not a white ball.

Your best claims about probabilities are always relative to what you know. For example, if all you know is that the die is a cube, then it's best to claim the probability of a 5 coming up on the next roll is 1/6. But if you also know the die is loaded so that a 4 never comes up, then you should claim the probability of a 5 is greater than 1/6.

We make all sorts of probability judgments without putting any numbers on those probabilities. Looking at a woman walking out of a parking garage, we correctly say it's more probable that she's a bank clerk than that she's a bank clerk from Florida even though we have no good idea what the probability number is. But if we noticed that she had just walked away from her car that has Florida license plates, then we’d say it’s more probable that she’s a bank clerk from Florida than that she’s a bank clerk not from Florida.

Exercise \(\PageIndex{1}\)

Is it more probable that she's a bank clerk from Florida than that she's poor and lives in Florida and works as a clerk in a bank?

Answer

Yes. It is more probable that she has two characteristics than that she has those two plus another one