If you’re a game-show watcher, you’ve probably wondered how well you’d do as a contestant on your favorite show. Maybe you’re the friend in...
If you’re a game-show watcher, you’ve probably wondered how well you’d do as a contestant on your favorite show. Maybe you’re the friend in the group who knows all the answers to Jeopardy, or maybe you’re the person confidently guessing wrong each time. Have you ever calculated the actual probability of you winning? The odds are quite slim!
Let’s look at the show Who Wants to Be a Millionaire, a game show that started in 1998 and was incredibly popular for about two decades. In case you’re unfamiliar with it, contestants get the opportunity to answer fifteen multiple-choice questions of increasing difficulty, with an ever-increasing monetary reward. If they answer all fifteen correctly, they win one million dollars. Between 1998 and 2020, only twelve players in the United States version won the top prize. The first five questions are easy (sometimes ridiculously so!), designed to help the players win a bit of money and relax. Then they start getting harder. In the original version, players have three lifelines they can use if they don’t know the answer to a question: phone a friend (a preselected friend who is waiting to receive the call), ask the audience, or 50/50, which removes two incorrect answer choices, leaving two behind. As soon as the player answers one question incorrectly, the game is over and the player leaves with an amount of money determined by what tier of question they were on.
David Patrick, a 1999 contestant on the show (who left with $64,000, in case you’re curious), wrote an entry on the website The Art of Problem-Solving that explored the probability of a random person winning a million dollars on the show. Patrick walks us through the possibilities: Maybe you randomly guess on all fifteen questions because you don’t know any of the answers. That gives you a one in one billion chance of winning the million-dollar prize. Not so great. If you know the answers to the first five, as most people do, you only have to randomly guess on ten questions, and now your odds of winning are one in a million. Not too bad! But still not great. As he puts it, “If you played WWTBAM every thirty minutes, twenty-four-hours-a-day, it would ‘only’ take you fifty-seven years, on average, to win the million.” Finally, Patrick calculates that if you get the first ten correct, as he did, randomly guessing on only the last five, you have a one in one thousand chance of winning. He then works in the lifelines (assuming you haven’t used any of them yet) and figures out that you have a one in twenty chance of winning. The odds are still against you, but not terribly so. If you can get through those first ten without using any of your lifelines, you have a decent shot at a million dollars!
The probability of something happening like winning a million dollars on a game show is found by dividing the number of favorable outcomes by the total possible outcomes. For WWTBAM, each question has four answer choices, so the probability of getting each question right is one in four (one favorable outcome, or correct answer choice, divided by four possible outcomes, or total answer choices). So then why isn’t the probability of winning a million dollars one in four? For multiple correct answers in a row, we need to calculate compound probability, or the odds that something (a correct answer) will occur more than once. It’s like flipping a coin: Each time you flip it, there is a fifty-fifty chance of it landing on heads, but there is a much smaller chance of it landing on heads ten times in a row (a little less than one in a thousand!).
Luck and the Lottery
LET’S USE DICE TO EXAMINE probability more. People talk about “lucky sevens” in dice, and many people claim seven as their lucky number. But why would seven be any luckier than any other number? What makes seven so special? If you look at possible outcomes from rolling two dice, you can see that seven is more likely than any other number to appear. These are all the possible rolls of two dice, with the number in red representing one die (let’s say Die A) and the number in blue representing the other (Die B):
Sum Ways to roll that sum
0 Not possible
1 Not possible
2 1+1
3 1+2 2+1
4 1+3 2+2 3+1
5 1+4 2+3 3+2 4+1
6 1+5 2+4 3+3 4+2 5+1
7 1+6 2+5 3+4 4+3 5+2 6+1
8 2+6 3+5 4+4 5+3 6+2
9 3+6 4+5 5+4 6+3
10 4+6 5+5 6+4
11 5+6 6+5
12 6+6
The total possible outcomes every single way the two dice could land—is thirty-six. That means the denominator of our fraction will always be thirty-six. For the numerator, we look at the number of favorable outcomes, or outcomes we are looking for. Let’s say we’re trying to figure out the probability of rolling a four. There are three possible ways to get a four. Three divided by thirty-six is one in twelve, or .08. Probabilities are usually given as percents, so we can say there is an 8 percent chance of rolling a four. Similar calculations yield a one in thirty-six, or about 3 percent, chance of rolling a two or a twelve. There are more ways (six) to make seven than any other number in the table, so the probability of rolling a seven is one in six, or nearly seventeen percent. It turns out sevens aren’t necessarily “lucky,” but just mathematically more likely.
Speaking of luck, millions of people every day try their hand at the lottery. It may be fun to dream about winning and being able to afford a beach house, college for your kids and grandkids, and early retirement, but pretty much everyone agrees that playing the lottery isn’t a worthwhile investment. Every lottery game has different odds of winning, depending on how many combinations of numbers are possible. According to Investopedia, the odds of winning any payout in Powerball are one in 24.9. That doesn’t seem too outrageous, right? Maybe you’re thinking it is worth playing!
That’s not the whole picture, though. Powerball isn’t an all-or-nothing game. If you win, it turns out you have excellent odds (more than nine in ten, or above 90 percent) of winning just $4. The lowest price of a Powerball ticket is $2, so in essence you have a very slim chance of profiting any more than $2. In fact, the chances of you winning the jackpot in Powerball are one in 292.2 million. In the language of probability, that is highly unlikely. Put into perspective, you are much more likely to be struck by lightning in a given year than to win the Powerball jackpot.
Weather Forecasts
Probability also plays a key role in weather forecasting. All weather forecasts are probabilistic, meaning they are based on predictions about what will probably, but not definitely, happen given the existing conditions. Depending on where you live, you’ve likely experienced a forecast that was overblown or completely missed. We’ve seen headlines like “Massive winter storm coming today!” that lead to empty store shelves and closed schools, only to have just a dusting of snow fall. If you experienced that (and your kids stayed home from school), you probably derided your local weather forecaster: How could they get it so wrong? We’ve all seen the other extreme as well—hurricanes that hit in a different place than they were predicted to, or massive storms that seemed to come out of nowhere, again leading us to question our forecasters’ abilities.
You may have even heard stories of weather forecasters who famously got their forecasts wrong. Perhaps you’ve heard the story of Michael Fish, the BBC news weatherman who will never live down his words on October 15th of 1987, shortly before Britain’s “Great Storm” hit. “Earlier on today,” he told viewers that day, “a woman rang the BBC and said she heard there was a hurricane on the way . . . Well, if you’re watching, don’t worry, there isn’t!” Technically, he was right: The storm was a cyclone, not a hurricane. But it caused eighteen deaths and widespread damage across the United Kingdom, with some calling it the strongest storm to hit the area in three hundred years. Poor Michael Fish’s life was miserable after that (partly because he refused to admit he was wrong). Newspaper headlines the day after reveal how blindsided people felt.
Luckily, weather forecasts have significantly improved since 1987, but they are still predictions based on what forecasters, aided by computer models, think will most likely happen.
Let’s look more closely at the kind of forecast we see daily: the chance of precipitation. You may look at the weather forecast for your town and see that there is an 80 percent chance of rain today. But what exactly does that mean? Does it mean that eighty percent of your town will see rain? Or that it will rain for 80 percent of the day? For most (but not all) forecasters, it doesn’t mean either of those. Precipitation forecasts, or PoPs (probability of precipitation), as they’re known in the industry, are one forecaster’s guess about one point in your area seeing precipitation (defined as greater than .01 inches of rain) that day. If your weather app says there is an 80 percent chance of rain, that means that somewhere in your area will probably (80 percent likely) see rain that day. If it doesn’t rain on your house, that doesn’t mean the forecast was wrong; remember there was still a 20 percent chance that there would be no rain. It’s also possible that it rained two hundred yards down the road.
When you’re looking at weather forecasts, it’s helpful to keep in mind some basic terminology around probabilities. When students learn about probability in school, they usually learn about it as a continuum, where 0 percent is impossible and 100 percent is certain.
This simple scale can help you plan your day. If the chance of rain is less than 50 percent, you might want to water your garden. If it’s more than 50 percent, maybe hold off (and carry an umbrella with you to work). Either way, though, remember that even a 99 percent chance of rain doesn’t mean it will definitely rain exactly where you are that day try to go easy on your weather forecasters!
Unfortunately for them, two top Hungarian meteorologists were fired in 2022 when it didn’t rain. In August of 2022, Hungary’s national meteorological service predicted severe storms for the country’s national holiday, Saint Stephen’s Day. Based on the forecast, the government decided to postpone celebratory fireworks. When it didn’t rain, people were outraged, and the head of the weather service, as well as her deputy, were fired. The Hungarian government claimed that these two were about to be fired anyway, but the story stands that the two lost their jobs because it didn’t rain.
Calculating Probability for Profit
Probability doesn’t just come into play in weather forecasts and rolling dice. Some people have made massive amounts of money based on knowing how to calculate probabilities. Have you ever played cards with a card counter? Or are you one yourself? You know the type: those who analyze and keep track of every move in a card game, mentally tallying which cards have been played and which ones are left to play. Card counting can be as simple as keeping track of the face cards in a game of Go Fish, or as complicated as the MIT Blackjack team that trained students in the 1990s to beat casinos. It can involve not just knowing which cards are left to play, but the probabilities of certain cards appearing. Once you know those probabilities, you can bet large sums of money and beat the casino . . . until they throw you out, that is.
That’s what happened to the team from MIT. While card counting isn’t illegal, casinos don’t like it because it makes them lose money. The saying is that the house always wins, but with a dedicated card counter, the house can lose. So who were these MIT students and how did they manage to beat the house?
The MIT Blackjack team formed in the early 1990s as a group of students who were good at math of course and interested in gambling. Advised by a successful card counter named Bill Kaplan (who had postponed his college experience at Harvard to go win money in Vegas), the team of students learned how to beat the odds at Blackjack by keeping track of the cards that had been played. This was a business venture for Kaplan, and a quite lucrative activity for many of the students, as they joined the high rollers clubs at casinos across the country. One by one, though, the students got kicked out of casinos or barred from playing Blackjack at those casinos. A private investigator eventually realized that they all came from MIT, and the game was over, so to speak. The team folded and Kaplan moved on to other ventures, though some players continued to make their money as Blackjack champs. But the story lives on: A bunch of data nerds won hundreds of thousands of dollars by using what they knew about probability.
Genetics and the Likelihood of Redheads
Let’s look at another field where probability is used all the time: genetics. If you took high school biology, you may remember Punnett squares. To a fourteen year old, Punnett squares seem like magical predictors that lend a glimpse into what geneticists do. In reality, Punnett squares are a visual tool to represent some standard probability calculations.
If you are a biological parent, you likely put some thought into what your child would look like before they came out of the womb. Will the baby have my curly brown hair? Will he or she have my wife’s green eyes? Will the baby be long and lean, like my side of the family, or on the shorter side, like my wife’s family? How about my weird ability to roll my tongue will the baby get that? Sometimes the kids come out looking just like you expected. Sometimes, though, they don’t; they have some trait that nobody on either side of the family had except for your great aunt Edna.
To understand more about genetics, we must delve deeper into probability. In the last chapter, we looked at the probability of an event occurring. That’s a simple calculation: the number of favorable outcomes divided by the number of total possible outcomes. Now let’s look at compound probability. That means we want to figure out the likelihood of two independent events happening. There’s a handy rule to help us here: the product rule.
The product rule tells us that, to calculate the probability of two independent events both happening, we multiply their two probabilities. Let’s say we want to figure out how likely it is that a coin lands on heads two times in a row. For each flip, the probability of it landing on heads is ½. For the probability of it landing on heads one time AND a second time, we multiply ½ by itself (for each flip) and get ¼. There is a one in four possibility (or 25 percent) that a coin will land on heads twice in a row.
Punnett Squares
Now for some basics on genetics: Every person inherits copies of genes from two biological parents. These genes determine all sorts of traits everything that makes up an individual, in essence. Let’s call these two genes A and a, meaning two different versions (alleles) of the same gene, one from the mother and one from the father. In genetics, the dominant allele the one that appears in the child is represented with a capital letter, and the recessive allele the one that the child carries but that doesn’t show up—is represented with a lowercase letter. For example, let’s imagine a person with black hair who also inherited a recessive gene for blond hair. This person is said to be heterozygous, meaning they inherited two different alleles (hence the hetero- prefix) for a trait. We’ll call the dominant gene that produces black hair uppercase A, and the recessive gene that produces blonde hair lowercase a. This is Parent 1 in the diagram below:
Parent 1 has a child with another individual, Parent 2, who also happens to have black hair and a recessive gene for blond hair. This person is represented by the left-hand column of the diagram:
Since every child gets alleles from both parents, we complete the table with the possible combinations of alleles that their children could get from them:
Each of the squares in the table represents the probability that the child will have that combination of alleles. Remember that the dominant allele represented by the capital A represents black hair, and if it appears in the square, the child will present with that trait. So that means there is a three in four chance, or 75 percent probability, that the child will have black hair, and maybe surprisingly only a one in four chance (25 percent) that the child will have blond hair.
This might seem second-nature to you. If you and your partner both have black hair, it makes sense that your child will probably have black hair. But let’s look at the chances of two parents having a red-headed child. Redheads are notoriously rare, but why? Here’s a handy visual from the website Let’s Talk Science of the possible Punnett squares for these two people
You can see that there are many more ways for the child not to have red hair than there are for the child to have red hair. Even if one parent is a redhead, the other parent has to be carrying a gene for red hair for the possibility of it to express itself in the child. If just one of you says, “My great-grandmother had red hair; I wonder if our kid will,” science can pretty much guarantee that your kid will not have red hair. If only one parent has red hair and the other one carries the gene, let’s say their mother and brother have red hair there is still only a fifty-fifty chance that the child will have red hair. The only way you can be sure your child will have red hair is if both parents have red hair.
Those small punnett squares may have been the extent of your high school biology education. But the idea of a Punnett square a visual representation of probabilities goes far beyond two heterozygotes. The biologist Gregor Mandel, known as the father of modern genetics, extensively studied peas in the nineteenth century to figure out how dominant and recessive traits work. In particular, he studied dihybrid crosses what happens when two organisms with two different traits reproduce. The two traits he examined were color (whether the peas were green or yellow) and wrinkliness (whether they were smooth or wrinkly). A larger Punnett square illustrates his findings
We can see that crossing a round yellow pea with a wrinkled green pea turns out to have only a one in sixteen chance that the offspring will be wrinkled and green.
Genetic Testing
You may be thinking that’s enough of a science lesson for now, and who really cares about peas anyway. The basic idea of Mendelian genetics is used all the time by biologists and geneticists to predict risks of different diseases. Certain diseases are much more common in particular races or ethnicities. For example, sickle cell anemia is much more common among people of sub-Saharan African, Middle Eastern, Asian, or South American descent. According to the National Institute of Health in the United States, “about 1 in every 365 Black or African American babies is born with SCA. About 1 in every 13 are born with SCT (sickle cell trait, meaning they carry the gene but have less severe disease). In contrast, only 1 in 333 white babies is born with SCT.”
There are also many diseases that are more common among people of Jewish descent. Remember that parents don’t necessarily need to have the disease to be a carrier of it. Most geneticists and obstetricians recommend to their Jewish patients particularly those with Ashkenazi heritage that they get a panel of genetic testing done, commonly known as the “Jewish panel.” This panel can tell them if one or both parents are carriers for certain diseases and the likelihood of their children presenting with each of these diseases.
Let’s look at Tay-Sachs disease, a truly terrible neurodegenerative disease that usually takes its victim’s life by the age of five. This disease, like several others, is much more prevalent in those of Ashkenazi Jewish or French-Canadian descent. Although the disease is rare, it still behooves future parents to find out if they are carriers of the disease. Why subject themselves to that? Because the likelihood of the child having it is so high and the disease so terrible that future parents would probably want to be fully informed. For Tay-Sachs to present, both parents need to be carriers. Let’s make a Punnett square for this scenario, where both parents are carriers but not presenting with the disease. We can say each of their genes is represented by Tt the dominant non-altered gene plus the recessive altered gene.
From the square, we can see that there is a one in four chance (the tt at the bottom right) that the child will have Tay-Sachs disease. That’s a pretty high chance for something so awful. There is also a two in four (or 50 percent) chance that the child will be a carrier for the disease, meaning that child will likely want their future partner to get tested before they someday have children. This likelihood is the same for each child that this couple has. It doesn’t mean that, if they have four children, one of them will have Tay-Sachs and two others will be carriers, but it means that each pregnancy independently carries this risk.
To leave this para on a lighter note: Remember that knowledge is power. The more you know about genetic risks of different diseases, the more informed your decisions will be and the better prepared you will be to handle life’s challenges. You’ll also be better able to explain why your child doesn’t have red hair, despite your great aunt Edna being a flaming redhead.
