Statistics Articles

### Figuring Out What Probability Means

Probabilities come in many different disguises. Some of the terms people use for probability are *chance*, *likelihood*, *odds*, *percentage*, and *proportion*. But the basic definition of *probability* is the long-term chance that a certain outcome will occur from some random process. A probability is a number between zero and one — a proportion, in other words. You can write it as a percentage, because people like to talk about probability as a percentage chance, or you can put it in the form of odds. The term "odds," however, isn't exactly the same as probability. *Odds* refers to the ratio of the denominator of a probability to the numerator of a probability. For example, if the probability of a horse winning a race is 50 percent (1/2), the odds of this horse winning are 2 to 1.

## Understanding the concept of chance

The term *chance* can take on many meanings. It can apply to an individual ("What are my chances of winning the lottery?"), or it can apply to a group ("The overall percentage of adults who get cancer is . . ."). You can signify a chance with a percent (80 percent), a proportion (0.80), or a word (such as "likely"). The bottom line of all probability terms is that they revolve around the idea of a long-term chance. When you're looking at a random process (and most occurrences in the world are the results of random processes for which the outcomes are never certain), you know that certain outcomes can happen, and you often weigh those outcomes in your mind. It all comes down to long-term chance; what's the chance that this or that outcome is going to occur in the long term (or over many individuals)?

If the chance of rain tomorrow is 30 percent, does that mean it won't rain because the chance is less than 50 percent? No. If the chance of rain is 30 percent, a meteorologist has looked at many days with similar conditions as tomorrow, and it rained on 30 percent of those days (and didn't rain the other 70 percent). So, a 30 percent chance for rain means only that it's unlikely to rain.

## Interpreting probabilities: Thinking large and long-term

You can interpret a probability as it applies to an individual or as it applies to a group. Because probabilities stand for long-term percentages, it may be easier to see how they apply to a group rather than to an individual. But sometimes one way makes more sense than the other, depending on the situation you face. The following sections outline ways to interpret probabilities as they apply to groups or individuals so you don't run into misinterpretation problems.

### Playing the instant lottery

Probabilities are based on long-term percentages (over thousands of trials), so when you apply them to a group, the group has to be large enough (the larger the better, but at least 1,500 or so items or individuals) for the probabilities to really apply. Here's an example where long-term interpretation makes sense in place of short-term interpretation. Suppose the chance of winning a prize in an instant lottery game is 1/10, or 10 percent. This probability means that in the long term (over thousands of tickets), 10 percent of all instant lottery tickets purchased for this game will win a prize, and 90 percent won't. It doesn't mean that if you buy 10 tickets, one of them will automatically win.

If you buy many sets of 10 tickets, on average, 10 percent of your tickets will win, but sometimes a group of 10 has multiple winners, and sometimes it has no winners. The winners are mixed up amongst the total population of tickets. If you buy exactly 10 tickets, each with a 10 percent chance of winning, you might expect a high chance of winning at least one prize. But the chance of you winning at least one prize with those 10 tickets is actually only 65 percent, and the chance of winning nothing is 35 percent.

### Pondering political affiliation

You can use the following example as an illustration of the limitation of probability — namely that actual probability often applies to the percentage of a large group. Suppose you know that 60 percent of the people in your community are Democrats, 30 percent are Republicans, and the remaining 10 percent are Independents or have another political affiliation. If you randomly select one person from your community, what's the chance the person is a Democrat? The chance is 60 percent. You can't say that the person is surely a Democrat because the chance is over 50 percent; the percentages just tell you that the person is more likely to be a Democrat. Of course, after you ask the person, he or she is either a Democrat or not; you can't be 60-percent Democrat.

## Seeing probability in everyday life

Probabilities affect the biggest and smallest decisions of people's lives. Pregnant women look at the probabilities of their babies having certain genetic disorders. Before you sign the papers to have surgery, doctors and nurses tell you about the chances that you'll have complications. And before you buy a vehicle, you can find out probabilities for almost every topic regarding that vehicle, including the chance of repairs becoming necessary, of the vehicle lasting a certain number of miles, or of you surviving a front-end crash or rollover (the latter depends on whether you wear a seatbelt — another fact based on probability).

Here are a couple of examples of probabilities that affect people's everyday lives:

- Distributing prescription medications in specially designed blister packages rather than in bottles may increase the likelihood that consumers will take the medication properly, a new study suggests.
* (Source: Ohio State University Research News, June 20, 2005)*

In other words, the probability of consumers taking their medications properly is higher if companies put the medications in the new packaging than it is when the companies put the medicines in bottles. You don't know what the probability of taking those medications correctly was originally or how much the probability increases with this new packaging, but you do know that according to this study, the packaging is having some effect.

- According to State Farm Insurance, the top three cities for auto theft in Ohio are Toledo (580.23 thefts per 100,000 vehicles), Columbus (558.19 per 100,000), and Dayton-Springfield (525.06 per 100,000).

The information in this example is given in terms of rate; the study recorded the number of cars stolen each year in various metropolitan areas of Ohio. Note that the study reports the information as the number of thefts per 100,000 vehicles. The researchers needed a fixed number of vehicles in order to be fair about the comparison. If the study used only the number of thefts, cities with more cars would always rank higher than cities with fewer cars.

How did the researchers get the specific numbers for this study? They took the actual number of thefts and divided it by the total number of vehicles to get a very small decimal value. They multiplied that value by 100,000 to get a number that's fair for comparison. To write the rates as probabilities, they simply divided them by 100,000 to put them back in decimal form. For Toledo, the probability of car theft is 580.23/100,000 = 0.0058023, or 0.58 percent; for Columbus, the probability of car theft is 0.0055819, or 0.56 percent; and for Dayton-Springfield, the probability is 0.0052506, or 0.53 percent.

Be sure to understand exactly what format people use to discuss or report a probability, and be sure that the format allows for a fair and equitable comparison.