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Math Help - find the probabilities

  1. #1
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    find the probabilities

    Alice and Bob each choose at random a number between zero and one. We assume a uniform probability law under which the probability law under which the probability of an event is proportional to its area. Consider the following events:

    A: The magnitude of the sum of the two numbers is greater than 2/3
    B: At lest one of the numbers is greater than 2/3
    C: The two numbers are equal
    D: Alice's number is greater than 2/3


    Find the following probabilities:

    a) Pr(A)
    b) Pr(B)
    c) Pr(A∩B)
    d) Pr(C)
    e) Pr(A∩D)



    ok for each part i guess i draw up a venn diagram? but i'm not entirely sure because the question is worded differently from the questions i've had before.

    Could anyone help me understand the line "We assume a uniform probability law under which the probability law under which the probability of an event is proportional to its area."

    Not looking for a cheat just a way to start it.
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  2. #2
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    I will help you with part b. It will be a model for the rest.
    In the unit square below, a point in the area shaded yellow will have at least one of its coordinates greater than \frac{2}{3}.
    Therefore the probability of event B is just that area divided by the total area.

    Then you model the other events.
    Attached Thumbnails Attached Thumbnails find the probabilities-untitled.gif  
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  3. #3
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    thanks for your help its much appreciated. Before i move on would i be right in saying Pr(B) = 5/9 ?
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  4. #4
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    Quote Originally Posted by djmccabie View Post
    thanks for your help its much appreciated. Before i move on would i be right in saying Pr(B) = 5/9 ?
    Correct.
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  5. #5
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    for some reason i just cant remember how to do Pr(sum of 2 numbers)>(a number). I'm thinking it should be Pr(alice∩Bob)>5/9 but this cant be right because i don't know the values for alice or bob :/ do i need to use binomial?
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  6. #6
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    Quote Originally Posted by djmccabie View Post
    for some reason i just cant remember how to do Pr(sum of 2 numbers)>(a number). I'm thinking it should be Pr(alice∩Bob)>5/9 but this cant be right because i don't know the values for alice or bob :/ do i need to use binomial?
    Graph the line a+b>\frac{2}{3}.
    Shade the correct area inside the square.
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  7. #7
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    I am trying to solve the same problem.
    I found some answers but I do not know if I am correct.

    a) 17/18
    b) 5/9
    c) 5/36
    d) 1/2
    e) 1/12

    Some of the answers are strange. Can anyone tell me if the answers are correct and if not, what answers are incorrect to think them again please?
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  8. #8
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    Did you read my first reply to this question?
    You will see the your part (a) in not correct.

    As for part (c) the answer will suprise you. Think in terms of area.
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  9. #9
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    I'm just not getting this
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  10. #10
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    is part a 7/9?
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  11. #11
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    also could (part b) = (part c)?
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  12. #12
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    Quote Originally Posted by djmccabie View Post
    I'm just not getting this
    Here is another try.
    The blue line is b=-a+\frac{2}{3}
    Event (A) is the yellow shaded area: a+b>\frac{2}{3}.
    That area is the probability.

    Note that event (B) is a subevent of (A).
    Thus A\cap B=B.
    Attached Thumbnails Attached Thumbnails find the probabilities-untitled.gif  
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  13. #13
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    Quote Originally Posted by djmccabie View Post
    also could (part b) = (part c)?
    Is there any area in event (D)?
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  14. #14
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    Quote Originally Posted by Plato View Post
    Note that event (B) is a subevent of (A).
    Thus A\cap B=B.

    doesn't this show that part(b) = part(c)

    ie Pr(B) = Pr(A∩B)


    I plotted the square the same as you did also and came out with 7/9 is this wrong?
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  15. #15
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    Quote Originally Posted by djmccabie View Post
    doesn't this show that part(b) = part(c)
    ie Pr(B) = Pr(A∩B)
    I plotted the square the same as you did also and came out with 7/9 is this wrong?
    Why do you ask.
    Yes it is correct. Trust yourself.


    But I am confused. Did you change (C) & (D)
    In the OP D=A\cap B so P(D)=P(B).

    C the two numbers are equal.
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