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Math Help - Expected Value

  1. #1
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    Expected Value

    A sack contains 20 products, where two are defective. Given that five products are chosen randomly, determine the expected number of defective products.

    Let P_k be the probability of there being k defective products out of the 5 chosen.

    P_k=\frac {_2C_k \cdot _{18}C_{5-k}}{_{20}C_5}

    \therefore E(X) = 0 \cdot \frac {_2C_0 \cdot _{18}C_5}{_{20}C_5} + 1 \cdot \frac {_2C_1 \cdot _{18}C_4}{_{20}C_5} + 2 \cdot \frac {_2C_2 \cdot _{18}C_3}{_{20}C_5}

    Why is it k \cdot ...?
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  2. #2
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    Quote Originally Posted by chengbin View Post
    A sack contains 20 products, where two are defective. Given that five products are chosen randomly, determine the expected number of defective products.

    Let P_k be the probability of there being k defective products out of the 5 chosen.

    P_k=\frac {_2C_k \cdot _{18}C_{5-k}}{_{20}C_5}

    \therefore E(X) = 0 \cdot \frac {_2C_0 \cdot _{18}C_5}{_{20}C_5} + 1 \cdot \frac {_2C_1 \cdot _{18}C_4}{_{20}C_5} + 2 \cdot \frac {_2C_2 \cdot _{18}C_3}{_{20}C_5}

    Why is it k \cdot ...?
    I don't really understand the trouble here .... k is the number of defectives in the chosen sample.
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  3. #3
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    Why is it k \cdot P_k?
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  4. #4
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    Quote Originally Posted by chengbin View Post
    Why is it k \cdot P_k?
    This follows directly from the definition of expected value.
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