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Math Help - Probability Notation

  1. #1
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    Probability Notation

    What is the difference between ⊆ and ⊂ ?

    I am currently using Probability and Statistics: Crash Course. It told what ∪ and ∩ meant, but not ⊂. Strange. I gather that the latter means "subset of", but I don't know what ⊆ means at all.

    It says that Theorem 1-1 is:
    "If A1 ⊂ A2, then
    P(A1) <= P(A2) and P(A2-A1) = P(A1) - P(A2)"

    The above is as written, double-checked (except that the numbers are in subscript, and <= is a single character).

    The first part sounds right, but what about the second part? If A1 is a subset of A2, then shouldn't P(A1) - P(A2) be a negative number?

    And what is with subtracting A1 from A2 in parenthesis? I didn't think events necessarily had mathematical values, though probabilities of events obviously do.
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  2. #2
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    Quote Originally Posted by Boris B View Post
    What is the difference between ⊆ and ⊂ ?

    I am currently using Probability and Statistics: Crash Course. It told what ∪ and ∩ meant, but not ⊂. Strange. I gather that the latter means "subset of", but I don't know what ⊆ means at all.

    It says that Theorem 1-1 is:
    "If A1 ⊂ A2, then
    P(A1) <= P(A2) and P(A2-A1) = P(A1) - P(A2)"

    The above is as written, double-checked (except that the numbers are in subscript, and <= is a single character).

    The first part sounds right, but what about the second part? If A1 is a subset of A2, then shouldn't P(A1) - P(A2) be a negative number?

    And what is with subtracting A1 from A2 in parenthesis? I didn't think events necessarily had mathematical values, though probabilities of events obviously do.
    Maybe a simple example will help on a six sided die
    The sample space is

    S=\left[ 1,2,3,4,5,6\right]

    Let A_1 be rolling an odd

    A_1=\left[ 1,3,5\right]

    let A_2 be rolling a one

    A_2=\left[ 1\right]

    so A_2 \subset A_1

    A_1-A_2=\left[ 3,5 \right]

    P(A_1-A_2)=\frac{2}{6}

    but

    P(A_1)-P(A_2)=\frac{3}{6}-\frac{1}{6}=\frac{2}{6}
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  3. #3
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    What you're saying makes sense....

    I think there is a typo in the book!

    Does the ⊆ notation mean "the two sets may be equivalent, or the first is a subset?"
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    Quote Originally Posted by Boris B View Post
    What is the difference between ⊆ and ⊂ ?

    I am currently using Probability and Statistics: Crash Course. It told what ∪ and ∩ meant, but not ⊂. Strange. I gather that the latter means "subset of", but I don't know what ⊆ means at all.

    It says that Theorem 1-1 is:
    "If A1 ⊂ A2, then
    P(A1) <= P(A2) and P(A2-A1) = P(A1) - P(A2)"

    The above is as written, double-checked (except that the numbers are in subscript, and <= is a single character).

    The first part sounds right, but what about the second part? If A1 is a subset of A2, then shouldn't P(A1) - P(A2) be a negative number?

    And what is with subtracting A1 from A2 in parenthesis? I didn't think events necessarily had mathematical values, though probabilities of events obviously do.
    Hi Boris,

    I think there is a misprint in your book and the theorem should read

    "If A1 ⊂ A2, then
    P(A1) <= P(A2) and P(A2-A1) = P(A2) - P(A1)"

    ⊂, as in A⊂B, means "is a subset of". I think some books use this symbol to mean A is a proper subset of B (meaning A is not equal to B), and they use the symbol ⊆, as in A ⊆ B, to mean that A is a subset of, or equal to, B, but I think this usage is not common. Most of my textbooks just use ⊂ to denote "is a subset of" so A ⊂ B includes the possibility that A=B.

    The minus in A-B, where A and B are sets, is the "set difference" operator. A-B is the set of all elements which are in A but not in B. Some texts use a different symbol, \, for this operation, as in A \ B.
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