Resolved: "A sample of 3 items is selected at..."

Nashorn0o

Open question

2022-08-19

A sample of 3 items is selected at random from a box containing 20 items of which 4 are defective. Find the expected number of defective items in the sample.

Answer & Explanation

Lina Watson

Beginner2022-08-20Added 9 answers

Define random variable X that marks the number of defective items in our sample. Since we choose 3 items for
our sample, random variable X can assume values 0,1,2,3. Let's find the probability of X=k for k some of the
numbers that are listed in previous sentence. In total, there exist $(\begin{matrix} 20 \\ 3 \end{matrix})$ ways to choose our sample. We can choose
k defective items on $(\begin{matrix} 4 \\ k \end{matrix}) * (\begin{matrix} 16 \\ 3 - k \end{matrix})$ ways. Hence
$P (X = k) = \frac{(\begin{matrix} 4 \\ k \end{matrix}) * (\begin{matrix} 16 \\ 3 - k \end{matrix})}{(\begin{matrix} 20 \\ 3 \end{matrix})}$
Usin the formula for expectation, we have that
$E (X) = \sum_{k = 0}^{3} k * P (X = k) = \sum_{k = 0}^{3} k * \frac{(\begin{matrix} 4 \\ k \end{matrix}) * (\begin{matrix} 16 \\ 3 - k \end{matrix})}{(\begin{matrix} 20 \\ 3 \end{matrix})}$
$= \frac{1}{(\begin{matrix} 20 \\ 3 \end{matrix})} [1 * (\begin{matrix} 4 \\ 1 \end{matrix}) * (\begin{matrix} 16 \\ 2 \end{matrix}) + 2 * (\begin{matrix} 4 \\ 2 \end{matrix}) * (\begin{matrix} 16 \\ 1 \end{matrix}) + 3 * (\begin{matrix} 4 \\ 3 \end{matrix}) * (\begin{matrix} 16 \\ 1 \end{matrix})]$
$= \frac{684}{(\begin{matrix} 20 \\ 3 \end{matrix})} = \frac{684}{1140} = 0.6$
Result:
$E (X) = 0.6$

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