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  • 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.

Probability is an area of mathematics that is everywhere.

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A2 TED-Ed probability question quiz guessing true false

【TED-Ed】What happens if you guess - Leigh Nataro

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    VoiceTube posted on 2013/10/24
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