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Conditional probability

  1. Dec 16, 2013 #1
    1. The problem statement, all variables and given/known data


    A study of texting and driving has found that 40% of all fatal auto accidents
    are attributed to texting drivers, 1% of all auto accidents are fatal, and
    drivers who text while driving are responsible for 20% of all accidents. Find
    the percentage of non-fatal accidents caused by drivers who do not text.

    2. Relevant equations



    3. The attempt at a solution

    Let T denote texting while driving and let F denote fatal accidents.

    P(F|T) = .40
    P(F) = .01
    P(T) = .2

    I guess we are trying to find
    p(F[itex]^{c}[/itex]|T[itex]^{c}[/itex]
    = (p(F[itex]^{c}[/itex][itex]\bigcap[/itex] T[itex]^{c}[/itex]) / p(T[itex]^{c}[/itex]

    We know p(F|T) = p(F[itex]\bigcap[/itex]T) / p(T) = 0.4 => p(F[itex]\bigcap[/itex]T) = 0.08

    Also p(F[itex]^{c}[/itex] [itex]\bigcap[/itex]T[itex]^{c}[/itex]) = 1 - p(F[itex]\bigcup[/itex]T)

    p(F[itex]\bigcup[/itex]T) = p(F) + p(T) - p(F [itex]\bigcap[/itex]T) = .01 + .2 - .08 = .13

    I am gonna stop here because when I start plugging in everything I have I wind up with the wrong answer. Is ther an assumption I have wrong or have interpreted, as usual, the problem wrong?
     
  2. jcsd
  3. Dec 16, 2013 #2

    mfb

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    That would be "For texting drivers, 40% of all accidents are fatal", which does not match the problem statement.
     
  4. Dec 16, 2013 #3
    So would it be p(F[itex]\bigcap[/itex]T) = .40?


    You got any hints?
     
    Last edited: Dec 16, 2013
  5. Dec 16, 2013 #4

    Dick

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    Now that reads "in 40% of all accidents the accident was fatal and the driver was texting". Still not what you want, given the probability of a fatal accident is only 0.01. Try again. Read these probability statement back in english. Here's a big hint. What does P(T|F) mean? State it in english.
     
  6. Dec 16, 2013 #5
    P(T|F) reads the probability of an accident being caused by texting, given that it was fatal?
     
  7. Dec 16, 2013 #6

    Dick

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    Ok, but, you should read it a little more literally. Nobody said anything about texting being the cause. It is just the probability that driver was texting given the accident was fatal. Now what's the value of that given the problem statement?
     
  8. Dec 17, 2013 #7
  9. Dec 17, 2013 #8

    HallsofIvy

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    Imagine 1000 accidents. "1% of all auto accidents are fatal" so there are 10 fatal accidents. "40% of all fatal auto accidents are attributed to texting drivers" so 4 of those fatal accidents are attributable to texting. "drivers who text while driving are responsible for 20% of all accidents" so 200 accidents are attributable to drivers who text.

    That is, out of 200 accidents attributable to drivers who text, 4 of them are fatal and 16 are not fatal.
     
  10. Dec 17, 2013 #9

    Dick

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    Yes, there's a big difference between P(T|F) and P(F|T). You might want to take another look at the expression you wrote for what you are trying to find.
     
  11. Dec 17, 2013 #10

    haruspex

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    Typo (?):
    ... and 196 are not fatal.
     
  12. Dec 18, 2013 #11

    HallsofIvy

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    Yes, thanks. Unfortunately, I can no longer edit it so I cannot pretend I didn't make that blunder!
     
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