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Math Help - Actuarial Question: Solve using Inclusion-Exclusion. Need A little help

  1. #1
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    Exclamation Actuarial Question: Solve using Inclusion-Exclusion. Need A little help

    Using Inclusion-Exclusion I need to solve this Actuarial problem:

    Problem 5: A Grocery store finds that:
    1. 30% of its customers buy bananas
    2. 22% of its customers buy soy milk
    3. 45% of its customers buy cow milk
    4. one third of the customers buy bananas also buy soy milk
    5. one half of the customers who don't buy bananas do buy cow milk
    6. none of the customers buy both soy and cow milk.

    What percentage of customers don't buy bananas or milk?


    This is what I have found so far is
    let b = bananas
    let s = soy milk
    let c = cow milk

    | b | = 30%
    | s | = 22%
    | c | = 45%

    | b ∩ s | = 10%
    | s ∩ c | = 0%
    | b ∩ s ∩ c | = 0%

    What I believe I need is |b ∩ c| which I am assuming I get from statement 5. I am unsure how to approach number five to receive what I need to complete this equation:

    | b U c U s | = | b | + | c | + | s | - | b ∩ s | - | s ∩ c | - | b ∩ c | + | b ∩ s ∩ c |

    then take what I get and subtract it from 100 for the percentage of customers that don't buy bananas and milk.

    Am I taking the wrong approach? If not can you steer me in the proper direction for portion I am missing. Thanks
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  2. #2
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    Re: Actuarial Question: Solve using Inclusion-Exclusion. Need A little help

    Hey maker1.

    You definitely have the right approach since you want to find P((b U c U s)^C) = 1 - P(b U c U s)
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  3. #3
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    Re: Actuarial Question: Solve using Inclusion-Exclusion. Need A little help

    Thank you. So the only portion I am missing is the |b ∩ c| which I have to get from statement 5. Statement 5 tells me that 35% of customers that don't buy bananas by cow milk. So how do I get |b ∩ c| from this? This is the portion I am struggling to find to complete the problem.
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  4. #4
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    Re: Actuarial Question: Solve using Inclusion-Exclusion. Need A little help

    Recall that that (b ∩ c ∩ a) U (b ∩ c ∩ a^c) = (b ∩ c).
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  5. #5
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    Re: Actuarial Question: Solve using Inclusion-Exclusion. Need A little help

    I will see if I can find that in my notes to recall thank you. Do you mean a to be s? And a^c to be s to the power of c?
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  6. #6
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    Re: Actuarial Question: Solve using Inclusion-Exclusion. Need A little help

    Is this correct? set | b U c | = | b | + | c | - | b ∩ c | which was 35 = 30 + 45 - | b ∩ c | and solved it algebraically for | b ∩ c |
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  7. #7
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    Re: Actuarial Question: Solve using Inclusion-Exclusion. Need A little help

    A^c means the complement of the set A with respect to the whole probability space. So P(A^c) = 1 - P(A).

    Yes that formula you used is correct with regards to b OR c in relation to |b|, |c| and |b AND c|.
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  8. #8
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    Re: Actuarial Question: Solve using Inclusion-Exclusion. Need A little help

    Okay and that is the proper way to achieve | b ∩ c | . Thanks so much for all of your help. Great first experience with this forum.
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