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Math Help - Probability and Stats Question!!!

  1. #1
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    Exclamation Probability and Stats Question!!!

    John, who is 7 years old, is mapping out his future. He thinks that with probability
    2/5 he might
    become an astronaut (event
    A), with probability 1/4 he might be a basketball star (event B), and with probability 1/2 he might be happily married and have lots of children (event C). He accepts that he
    can’t be an astronaut and a basketball star at the same time but thinks that astronauts and basketball
    stars are just as likely as anyone else to have lots of children.
    (i) Draw a Venn diagram illustrating this situation.
    (ii) Calculate the probability that John has lots of children but does not get to be an astronaut
    or a basketball star.
    (iii) What is the probability that none of John’s dreams comes true?
    [6 marks]

    4.
    A test for a disease has probability 0.96 of giving a positive result when the disease is in fact present,
    and probability 0.99 of giving a negative result when the disease is absent.
    (i) About 1 in 200 adults have the disease. If a randomly selected adult is tested for the disease,
    what is the probability of a positive result?
    (ii) A doctor believes there is a 25% chance that his patient has the disease. The patient is tested
    and the test comes up positive. What is now the probability that the patient has the disease?

    [6 marks]

    Help On either of these questions would be great!!!
    Thanks
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  2. #2
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    Quote Originally Posted by agrabham View Post
    John, who is 7 years old, is mapping out his future. He thinks that with probability
    2/5 he might

    become an astronaut (event A), with probability 1/4 he might be a basketball star (event B), and with probability 1/2 he might be happily married and have lots of children (event C). He accepts that he
    canít be an astronaut and a basketball star at the same time but thinks that astronauts and basketball
    stars are just as likely as anyone else to have lots of children.
    (i) Draw a Venn diagram illustrating this situation.
    (ii) Calculate the probability that John has lots of children but does not get to be an astronaut
    or a basketball star.
    (iii) What is the probability that none of Johnís dreams comes true?
    [6 marks]

    [snip]
    There's not enough information for a unique answer IF the to (iii) is assumed to be non-zero. To draw the Venn diagram, draw a circle labelled C and let it intersect with circles labelled A and B. The circles labelled A and B do not intersect. Let Pr(A and C') = a. It looks like you can assume that Pr( A and C) = Pr(A and B) = b, say. Let Pr(C and A' and B') = c. Let Pr(B and C') = d. Assuming the answer to (iii) is zero then:

    a + b = 2/5 .... (1)
    b + d = 1/4 .... (2)
    2b + c = 1/2 .... (3)
    a + 2b + c + d = 1 .... (4)

    I get a = 0.325, b = 0.075, c = 0.35, d = 0.175.
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  3. #3
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    Quote Originally Posted by agrabham View Post
    [snip]
    4. A test for a disease has probability 0.96 of giving a positive result when the disease is in fact present,
    and probability 0.99 of giving a negative result when the disease is absent.
    (i) About 1 in 200 adults have the disease. If a randomly selected adult is tested for the disease,
    what is the probability of a positive result?
    (ii) A doctor believes there is a 25% chance that his patient has the disease. The patient is tested
    and the test comes up positive. What is now the probability that the patient has the disease?
    [6 marks]

    Help On either of these questions would be great!!!
    Thanks
    Draw a tree diagram. Start with branches for D and D'. Then for each, have a branch for +ve and -ve. Note that Pr(D) = 1/200 = 0.005. Therefore Pr(D') = 1 - 0.005 = 0.995.

    (i) From the tree diagram I get Pr(+ve) = (0.005)(0.96) + (0.995)(0.01) = ......


    (ii) Edit the tree diagram by:

    Replace Pr(D) = 0.005 with Pr(D) = 0.25.
    Replace Pr(D') = 0.995 with Pr(D') = 0.75.

    Pr(D | +ve) = Pr(D and +ve)/Pr(+ve).

    From the tree diagram I get Pr(D | +ve) = (0.25)(0.96)/[(0.25)(0.96) + (0.75)(0.01)] = ....
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