Expected Value

**chengbin** · September 24th 2009, 06:43 PM

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 $k \cdot ...$ ?

**mr fantastic** · September 24th 2009, 10:19 PM

Originally Posted by chengbin

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.

**chengbin** · September 25th 2009, 03:09 AM

Why is it $k \cdot P_k$ ?

**mr fantastic** · September 25th 2009, 03:14 AM

Originally Posted by chengbin

Why is it $k \cdot P_k$ ?

This follows directly from the definition of expected value.

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