Learn what probability is and how it affects business, baseball, weather forecasts, and even the color of your eyes.
Heads or tails? Do you have a favorite side that you rely on in coin tosses? Does it matter which one you choose? If you toss a coin 4 times and get 4 heads in a row, are the chances good that the 5th toss will come up tails? What’s the math behind all of this? And what does all of this coin flipping have to do with business, baseball, and weather forecasts? Stay tuned because these are exactly the questions we’ll be answering over the next few weeks as we tackle the exciting subject of probability.
What Is Probability?
Before we get too far into the math of probability, let’s take a minute to think about what probabilities are in the first place. In particular, let’s think about batting averages in baseball. If you’re not familiar with batting averages, they’re a decimal number between 0 and 1 that tell you something about how good a hitter is. A value of 0.300 (which people usually call “batting 300”) is very good. But what exactly does that mean?
Batting averages are calculated by dividing the total number of hits a player has by the total number of times they’ve batted (actually, it’s the total number of at bats minus the number of times they’ve walked or been hit by a pitch). As you can see, this is just an average—actually the mean—hence the name “batting average.” In this case, 0.300 indicates that the player has hit safely 30% of the time. Which means that when the player next comes to bat, there is a probability of 0.3—or a 30% chance—that they’ll get a hit. This same type of probabilistic thinking can be applied to help us understand things like the probability that a stock will increase in value, the probability that it will rain tomorrow, and the probabilities behind the genes you inherited that determined the color of your eyes.
How Is Probability Defined?
Just like batting averages, probabilities are decimal numbers between 0 and 1. The higher the probability (meaning the closer to 1), the more likely it is that whatever we’re talking about will actually happen. If we ignore getting walked or hit by a pitch in our baseball example, there are two things that can happen when a batter comes to the plate: (1) they can get a hit, or (2) they can make an out. As we’ve seen, the player’s batting average tells us the probability that they’ll get a hit. So what’s the probability that they’ll make an out? Well, since we know that there are two total possibilities, the probabilities of each of these two things must add up to 1 since we’re certain that one or the other has to happen. Which means that: P(out) = 1 – P(hit). In other words, if a player has a batting average of 0.300, they have a “getting out average” of 1 – 0.300 = 0.700. The quick and dirty rule to remember is that the sum of the probabilities of all the different things that can happen must add up to a total of 1.
What’s the Probability of Tossing Heads?
Now that we’re up to speed on the basics of probability, let’s take a look at the classic case of a coin toss. As we all know, there are two possibilities when you flip a coin: heads or tails (we’re going to ignore the extremely tiny chance that the coin could land on its side). And the likelihood of each outcome is the same…there’s a 50/50 chance—in other words a probability of 1/2—of coming up either heads or tails.
Or is there? Although we normally don’t think about this, there’s really no way to look at a coin and say with certainty that heads and tails are equally likely outcomes. So why do we make this assumption? Well, the only practical way to test it is to take the coin and flip it a huge number of times—a million or so would be good—to check if the tosses come up heads and tails fairly evenly over the long haul. And I don’t know about you, but that doesn’t sound like a lot of fun! Which is why we rely on the fact that enough people throughout history have done this little experiment for us and have determined that normal coins are indeed what we call “fair coins”—meaning the probability of heads and tails are both very close to 1/2. As you can check, the sum of these two probabilities is 1/2 + 1/2 = 1, exactly as it must be since one of the two outcomes always happens.
Beware the Gambler’s Fallacy!
Let’s now imagine that a friend has challenged you to a very simple game of chance. With each toss of a coin, you have to guess “heads or tails.” Answer right…you win. Get it wrong…you lose. (I know—pretty high stakes!) Since you’re a probability expert, you know that the probability of the coin landing heads side up is 1/2 every toss. Because it therefore doesn’t really matter what you guess, you decide to go with heads every toss. And that choice is going swimmingly well as the first four tosses have all come up heads! But that gets you thinking: After four consecutive heads, aren’t the odds really good that the next toss will be tails? After all, isn’t the coin due to come up tails at this point? What do you think? Should you switch your guess to tails? Or stick with heads instead?
Unfortunately, that’s all the math we have time for today. Which means that the answer to this question—which, by the way, is a version of a something known as the Gambler’s Fallacy (the name of which gives a clue to the answer)—will have to wait until next time when we pick up the story and talk about how to actually get in there and calculate probabilities in more complex situations.
In the meantime, remember to become a fan of the Math Dude on Facebook where you’ll find lots of great math tips posted throughout the week. If you’re on Twitter, please follow me there, too. Finally, please send your math questions my way via Facebook, Twitter, or email at email@example.com.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Image courtesy of Shutterstock