What happens if you guess - Leigh Nataro

Probability is an area of mathematics that is everywhere. We hear about it in weather forecasts, like there's an 80% chance of snow tomorrow. It's used in making predictions in sports, such as determining the odds for who will win the Super Bowl. Probability is also used in helping to set auto insurance rates, and it's what keeps casinos and lotteries in business.

How can probability affect you? Let's look at a simple probability problem. Does it pay to randomly guess on all 10 questions on a true/false quiz? In other words, if you were to toss a fair coin 10 times and use it to choose the answers, what is the probability you would get a perfect score? It seems simple enough. There are only two possible outcomes for each question. But with a 10-question true/false quiz, there are lots of possible ways to write down different combinations of Ts and Fs.

To understand how many different combinations, let's think about a much smaller true/false quiz with only two questions. You could answer "true true," or "false false," or one of each—first "false" then "true," or first "true" then "false." So that's four different ways to write the answers for a two-question quiz. What about a 10-question quiz? Well, this time, there are too many to count and list by hand.

In order to answer this question, we need to know the fundamental counting principle. The fundamental counting principle states that if there are A possible outcomes for one event and B possible outcomes for another event, then there are A times B ways to pair the outcomes. Clearly, this works for a two-question true/false quiz. There are two different answers you could write for the first question and two different answers you could write for the second question. That makes 2 times 2, or 4 different ways to write the answers for a two-question quiz.

Now let's consider the 10-question quiz. To do this, we just need to extend the fundamental counting principle a bit. We need to realize that there are two possible answers for each of the 10 questions. So the number of possible outcomes is 2 times 2 times 2 times 2 times 2 times 2 times 2 times 2 times 2 times 2. Or, a shorter way to say that is 2 to the 10th power, which is equal to 1,024. That means of all the ways you could write down your Ts and Fs, only one of the 1,024 ways would match the teacher's answer key perfectly.

So the probability of you getting a perfect score by guessing is only 1 out of 1,024, or about a 10th of a percent. Clearly, guessing isn't a good idea. In fact, what would be the most common score if you and all your friends were to always randomly guess at every question on a 10-question true/false quiz? Well, not everyone would get exactly 5 out of 10. But the average score, in the long run, would be 5.

In a situation like this, there are two possible outcomes: a question is right or wrong, and the probability of being right by guessing is always the same: 1/2. To find the average number you would get right by guessing, you multiply the number of questions by the probability of getting the question right. Here, that is 10 times 1/2, or 5. Hopefully, you study for quizzes, since it clearly doesn't pay to guess.

But at one point, you probably took a standardized test like the SAT, and most people have to guess on a few questions. If there are 20 questions and five possible answers for each question, what is the probability you would get all 20 right by randomly guessing? And what should you expect your score to be? Let's use the ideas from before. First, since the probability of getting a question right by guessing is 1/5, we would expect to get 1/5 of the 20 questions right. Yikes—that's only four questions!

Are you thinking that the probability of getting all 20 questions correct is pretty small? Let's find out just how small. Do you recall the fundamental counting principle that was stated before? With five possible outcomes for each question, we would multiply 5 times 5 times 5 times 5 times... Well, we would just use 5 as a factor 20 times, and 5 to the 20th power is 95 trillion, 365 billion, 431 million, 648 thousand, 625. Wow—that's huge! So the probability of getting all questions correct by randomly guessing is about 1 in 95 trillion.