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 (see the previous section), 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.

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).

While scanning the Internet, I quickly found several examples of probabilities that affect people’s everyday lives — two of which I list here:

• 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 , 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.

Be subjective

The subjective approach to probability is the most vague and the least scientific. It’s based mostly on opinions, feelings, or hopes, meaning that you typically don’t use this type of probability approach in real scientific endeavors. You basically say, “Here’s what I think the probability is.” For example, although the actual, true probability that the Ohio State football team will win the national championship is out there somewhere, no one knows what it is, even though every fan and analyst will have ideas about what that chance is, based on everything from dreams they had last night, to how much they love or hate Ohio State, to all the statistics from Ohio State football over the last 100 years. Other people will take a slightly more scientific approach — evaluating players’ stats, looking at the strength of the competition, and so on. But in the end, the probability of an event like this is mostly subjective, and although this approach isn’t scientific, it sure makes for some great sports talk amongst the fans!

Take a classical approach

The classical approach to probability is a mathematical, formula-based approach. You can use math and counting rules to calculate exact probabilities in many cases (for more on the counting rules, see Chapter 5). Anytime you have a situation where you can enumerate the possible outcomes and figure their individual probabilities by using math, you can use the classical approach to getting the probability of an outcome or series of outcomes from a random process.

For example, when you roll two die, you have six possible outcomes for the first die, and for each of those outcomes, you have another six possible outcomes for the second die. All together, you have possible outcomes for the pair. In order to get a sum of two on a roll, you have to roll two 1s, meaning it can happen in only one way. So, the probability of getting a sum of two is . The probability of getting a sum of three is , because only two of the outcomes result in a sum of three: 1-2 or 2-1. A sum of seven has a probability of , or — the highest probability of any sum of two die. Why is seven the sum with highest probability? Because it has the most possible ways of coming up: 1-6, 2-5, 3-4, 4-3, 5-2, and 6-1. That’s why the number seven is so important in the gambling game craps. (For more on this example, see Chapter 2.)

You also use the classical approach when you make certain assumptions about a random process that’s occurring. For example, if you can assume that the probability of achieving success when you’re trying to make a sale is the same on each trial, you can use the binomial probability model for figuring out the probability of making 5 sales in 20 tries. Many types of probability models are available, and I discuss many of them in this book. (For more on the binomial probability model, see Chapter 8.)

The classical approach doesn’t work when you can’t describe the possible individual outcomes and come up with some mathematical way of determining the probabilities. For example, if you have to decide between different brands of refrigerators to buy, and your criterion is having the least chance of needing repairs in the next five years, the classical approach can’t help you for a couple reasons. First, you can’t assume that the probability of a refrigerator needing one repair is the same as the probability of needing two, three, or four repairs in five years. Second, you have no math formula to figure out the chances of repairs for different brands of refrigerators; it depends on past data that’s been collected regarding repairs.

Find relative frequencies

In cases where you can’t come up with a mathematical formula or model to figure a probability, the relative frequency approach is your best bet. The approach is based on collecting data and, based on that data, finding the percentage of time that an event occurred. The percentage you find is the relative frequency of that event — the number of times the event occurred divided by the total number of observations made. (You can find the probabilities for the refrigerator repairs example in the previous section with the relative frequency approach by collecting data on refrigerator repair records.)

Suppose, for example, that you’re watching your birdfeeder, and you notice a lot of cardinals coming for dinner. You want to find the probability that the next bird that comes to the feeder is a cardinal. You can estimate this probability by counting the number of birds that come to your feeder over a period of time and noting how many cardinals you see. If you count 100 bird visits, and 27 of the visitors are cardinals, you can say that for the period of time you observe, 27 out of 100 visits — or 27 percent, the relative frequency — were made by cardinals. Now, if you have to guess the probability that the next bird to visit is a cardinal, 27 percent would be your best guess. You come up with a probability based on relative frequency.

A limitation of the relative frequency approach is that the probabilities you come up with are only estimates because you base them on finite samples of data you collect. And those estimates are only as good as the data that you collect. For example, if you collected your birdfeeder data when you offered sunflower seeds, but now you offer thistle seed (loved by smaller birds), your probability of seeing a cardinal changes. Also, if you look at the feeder only at 5 p.m. each day, when cardinals are more likely to be out than any other bird, your predictions work only at that same time period, not at noon when all the finches are also out and about. The issue of collecting good data is a statistical one; see Statistics For Dummies (Wiley) for more information.

Use simulations

The simulation approach is a process that creates data by setting up a certain scenario, playing out that scenario over and over many times, and looking at the percentage of times a certain outcome occurs. It may sound like the relative frequency approach (see the previous section), but it’s different in three ways:

• You create the data (usually with a computer); you don’t collect it out in the real world.
• The amount of data is typically much larger than the amount you could observe in real life.
• You use a certain model that scientists come up with, and models have assumptions.

You can see an example of a simulation if you let a computer play out a game of chance for you. You can tell it to credit you with $1.00 if a head comes up on a coin flip and deduct $1.00 if a tail comes up. Repeat the bet thousands of times and see what you end up with. Change the probabilities of heads and tails to see what happens. Your experiments are examples of simple simulations.

One commonality between simulations and the relative frequency approach is that your results are only as good as the data you come up with. I remember very clearly a simulation that a student performed to predict which team would win the NCAA basketball tournament some years ago. The student gave each of the 64 teams in the tournament a probability of winning its game based on certain statistics that the sports gurus came up with. The student fed those probabilities into the computer and made the computer repeat the tournament over and over millions of times, recording who won each game and who won the entire tournament. On 96 percent of the simulations, Duke University won the whole thing. So, of course, it seemed as if Duke was a shoe-in that year. Guess how long Duke actually lasted? The team went down in the second of six rounds.