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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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