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What happens if you guess - Leigh Nataro

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    Probability is an area of mathematics
    that is everywhere.
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    We hear about it in weather forecasts,
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    like there's an 80% chance
    of snow tomorrow.
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    It's used in making predictions in sports,
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    such as determining the odds
    for who will win the Super Bowl.
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    Probability is also used in helping
    to set auto insurance rates
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    and it's what keeps casinos
    and lotteries in business.
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    How can probability affect you?
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    Let's look at a simple
    probability problem.
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    Does it pay to randomly guess
    on all 10 questions
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    on a true/ false quiz?
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    In other words,
    if you were to toss a fair coin
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    10 times, and use it
    to choose the answers,
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    what is the probability
    you would get a perfect score?
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    It seems simple enough. There are only two
    possible outcomes for each question.
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    But with a 10-question true/ false quiz,
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    there are lots of possible ways
    to write down different combinations
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    of Ts and Fs. To understand
    how many different combinations,
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    let's think about a much smaller
    true/ false quiz
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    with only two questions.
    You could answer
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    "true true," or "false false,"
    or one of each.
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    First "false" then "true,"
    or first "true" then "false."
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    So that's four different ways to write
    the answers for a two-question quiz.
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    What about a 10-question quiz?
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    Well, this time, there are too many
    to count and list by hand.
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    In order to answer this question, we need
    to know the fundamental counting principle.
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    The fundamental counting principle states
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    that if there are A possible outcomes
    for one event,
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    and B possible outcomes for another event,
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    then there are A times B ways
    to pair the outcomes.
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    Clearly this works
    for a two-question true/ false quiz.
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    There are two different answers
    you could write for the first question,
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    and two different answers you could
    write for the second question.
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    That makes 2 times 2, or, 4 different ways
    to write the answers for a two-question quiz.
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    Now let's consider the 10-question quiz.
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    To do this, we just need to extend
    the fundamental counting principle a bit.
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    We need to realize that there are two
    possible answers for each of the 10 questions.
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    So the number of possible outcomes is
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    2, times 2, times 2, times 2,
    times 2, times 2,
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    times 2, times 2, times 2, times 2.
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    Or, a shorter way to say
    that is 2 to the 10th power,
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    which is equal to 1,024.
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    That means of all the ways
    you could write down your Ts and Fs,
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    only one of the 1,024 ways would match
    the teacher's answer key perfectly.
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    So the probability of you getting
    a perfect score by guessing
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    is only 1 out of 1,024,
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    or about a 10th of a percent.
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    Clearly, guessing isn't a good idea.
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    In fact, what would be
    the most common score
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    if you and all your friends
    were to always randomly guess
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    at every question on
    a 10-question true/ false quiz?
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    Well, not everyone would get
    exactly 5 out of 10.
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    But the average score, in the long run,
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    would be 5.
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    In a situation like this,
    there are two possible outcomes:
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    a question is right or wrong,
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    and the probability
    of being right by guessing
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    is always the same: 1/2.
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    To find the average number
    you would get right by guessing,
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    you multiply the number of questions
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    by the probability
    of getting the question right.
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    Here, that is 10 times 1/2, or 5.
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    Hopefully you study for quizzes,
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    since it clearly doesn't pay to guess.
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    But at one point, you probably took
    a standardized test like the SAT,
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    and most people have to guess
    on a few questions.
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    If there are 20 questions
    and five possible answers
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    for each question, what is the probability
    you would get all 20 right
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    by randomly guessing?
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    And what should you expect
    your score to be?
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    Let's use the ideas from before.
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    First, since the probability of getting
    a question right by guessing is 1/5,
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    we would expect to get 1/5
    of the 20 questions right.
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    Yikes - that's only four questions!
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    Are you thinking that the probability
    of getting all 20 questions correct is pretty small?
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    Let's find out just how small.
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    Do you recall the fundamental
    counting principle that was stated before?
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    With five possible outcomes
    for each question,
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    we would multiply 5 times 5
    times 5 times 5 times...
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    Well, we would just use 5 as a factor
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    20 times, and 5 to the 20th power
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    is 95 trillion, 365 billion, 431 million,
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    648 thousand, 625.
    Wow - that's huge!
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    So the probability of getting all questions
    correct by randomly guessing
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    is about 1 in 95 trillion.
Title:
What happens if you guess - Leigh Nataro
Description:

View full lesson: http://ed.ted.com/lessons/leigh-nataro-what-happens-if-you-guess

Will it rain tomorrow? How likely is your favorite team to win the Super Bowl? Questions like these are answered through the mathematics of probability. Watch this artistic visualization of your odds of passing a test if you don't know any of the answers.

Lesson by Leigh Nataro, animation by Matthew Saunders.
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Video Language:
English
Team:
closed TED
Project:
TED-Ed
Duration:
05:28
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