Percentages aren’t the same as random chance

This is a follow-up post to my More likely to die how? post from four years ago, because I still keep hearing people equating percentages with chance.

Look, people shouldn’t get flipped out about whether or not their marriage is going to fail or not. You don’t have a 50/50 chance of getting a divorce because 50% of marriages end in divorce. There are about 50% of married couples who have a 95% chance of getting divorced and about 50% of married couples who have a 95% chance of staying married. If you believe that marriage is about feelings and always being infatuated with each other, then your marriage is going to fail. If you believe that marriage is about commitment and staying together through thick and thin (or “for better or for worse,” as the vow typically goes), then your marriage is going to work.

Think about it another way. Let’s say your high school has a graduation rate of 75%. That means 1 out of every 4 students drops out, fails out, or gets kicked out before completing 12th grade (yes, I’m writing from an American context—I’m American, so humour me). It also means 3 out of every 4 students completes 12th grade and graduates. Does that mean every single student has a 1 in 4 chance of not graduating? Absolutely not. The kids in the honors classes who don’t suffer from extreme perfectionism or depression, who do their homework, and who don’t skip class have almost a 100% chance of graduating, assuming they don’t get hit by a bus the day before graduation. The kids who never complete assignments, skip school, and have chronic behavioral problems have almost 100% chance of not graduating.

Likewise, again I keep hearing about how planes are much safer to travel in than cars. Why are there variable insurance rates, depending on gender, age, how long you’ve been driving, etc.? If everyone has an equal chance of getting into a fatal car accident, wouldn’t insurance companies have the same rates for everybody? But they don’t. They know a 16-year-old male who just got his license is far more likely to get into an accident than a 36-year-old female who has been driving for twenty years. There are exceptions, of course. Some 16-year-old males are both skillful and cautious, and some people who have been driving for decades are both unskillful and reckless, despite their experience. If I get into an airplane, no matter how “skilled” I am as a passenger, I have the same chance of dying in a plane crash as all the other passengers, because we have no control over the plane’s operations, maintenance, pilot qualifications, etc. We have no control whatsoever. So whether you die in a plane crash or not is 100% up to chance. Whether you die in a car crash or not is only partially up to chance. If you’re skillful and cautious, you’re far less likely to get into a car accident. You can’t be 100% sure you’ll avoid all crashes. After all, someone may rear-end you. A drunk driver might run a red light and smash right into you. But you will avoid many accidents that other less skillful or more reckless drivers will still have to face.

Percentages are the same as chance only if you have no influence on the outcome. Your academic work influences your chances of graduating high school. Your commitment to marriage influences your chances of staying married. Your skill and caution in driving influences your chances of avoiding a car accident. There are some things you have control over and others you don’t, but either way, you may still end up with the same percentages. Percentages are not always chance.

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6 Comments

  1. While I agree with your reasoning, your math just doesn’t work. Saying “50% of marriages end in divorce” is the EXACT SAME THING as saying, “Take a marriage at random. There is a 1/2 chance it will fail.” Yes, you can try to work against these odds by keeping a positive outlook, marrying for the right reasons, etc, but that does not change the cold, hard facts: 50% of marriages fail.

    Similarly with the high school graduation rates. On September 1 (first day of school), say there are 200 incoming freshmen. 50 of them will not graduate. Which 50, of course, is dependent upon various mitigating factors, but taken one-at-at-time, and viewed equally, each of the 200 has a 1/4 chance of not graduating.

    This is just math, not philosophy. Mathematics says that 1/4 == 75%. From wikipedia, “In mathematics, a percentage is a way of expressing a number as a fraction of 100 (per cent meaning “per hundred”).” [http://en.wikipedia.org/wiki/Percentages] So, “75%” and “1/4” are two ways of writing the same thing.

    Now, I actually agree with you, and I think that Statistics DO lie. But I’m just saying: the terminology is not something we can argue with. It’s all in the analysis.

  2. All you’re really saying is that 75% = 3/4

    Yes, that’s simple math.

    But saying that any given student has the same chances of graduating as any other given student is bollocks. Think about it.

    Let’s say there are two couples, Britney Spears-Kevin Federline and my parents. Yes, out of those two couples, one couple ended in divorce; another didn’t. But Britney didn’t have a 50/50 chance of divorcing, and neither did my parents. The results (Britney’s divorce, my parents’ decades of continuous marriage) were not surprising at all.

    Likewise, every race will have a winner and a loser. If Alberto Salazar and I are in the same race, we do not stand an equal chance of winning or losing (I’ll leave it up to you to guess who’d win), even though only 1 out of number-of-people-racing wins any given race (with the exception of a tie).

    In other words, as I tried to explain in the original post (but failed to convey properly, apparently), the aggregate probability for a population does not apply equally to all members of that population if the members’ skill or knowledge can affect the outcome of what’s being measured.

    Let’s say 1 out of 10 people can successfully install and configure Ubuntu on a previously Windows-preloaded computer. That means that if you don’t know anything about those people, you have a 1 in 10 chance of guessing which one is the one who can successfully install and configure Ubuntu. It does not mean that every single one of those 10 individuals has a 1 in 10 chance of successfully installing and configuring Ubuntu. The one user who knows how to repartition drives, set the BIOS to boot, and configure the /etc/X11/xorg.conf file does not have the same chance of succeeding as the one user who has no idea how to boot a CD or what an operating system is.

    This is different from situations in which participants have no influence on the outcome. For example, in Blackjack, you are dealt the cards you are dealt. Everyone, regardless of their skill at bluffing or knowing what to bet, has an equal chance of getting (or not getting) 21 in the first two cards. The cards are randomly dealt, so if you select a person at random, she will have the same chances of getting 21 on the first two cards as any other person you select at random.

  3. math is a wonderful thing, however, stats have to match population. if ten people stand in a room and one of them can install ubuntu, that does not mean that each person has a 1/10 chance or 10% chance to install the program! it simply means that 1 in those ten persons has the capability to install the program.

    statistics are based on a population coefficient from any given probability equation.

    for example: i pass all of my math tests by 10 pts, a 75, average. this does not give me a 25% chance of failure. this means that if i were to be tested at any given time, i would get approximately 75% of the material correct.

    one must understand what the probability is compared to, to understand the statistic.

    also, the chance of a teen getting in a car crash are much higher than an adult, based on average statistics by age group, which are supported by results of experience. you do of course have the same chance of surviving a flight as the passenger next to you, but you also share this percentage with the pilot. this percentage is and must be represented by all variables including pilot experience and plane malfunctions.

    all said, one must understand what a statistic is applied to before they can understand the meaning of the stat.

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