A sample of 3 items is selected at random from a box containing 20 items of which 4 are defective

# A sample of 3 items is selected at random from a box containing 20 items of which 4 are defective

H
240 points

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.

A sample
hansel

8.7k points

Find the different probabilities for picking {0,1,2,3} defective items,

let X be the number of defective items chosen,

$$P(X=0)=\frac{C(16,3)C(4,0)}{C(20,3)}=0.491$$

$$P(X=1)=\frac{C(16,2)C(4,1)}{C(20,3)}=0.421$$

$$P(X=2)=\frac{C(16,1)C(4,2)}{C(20,3)}=0.084$$

$$P(X=3)=\frac{C(16,0)C(4,3)}{C(20,3)}=0.0035$$

therefore the expected number of defective items is given by,

$$E(X)=0(0.491)+1(0.421)+2(0.084)+3(0.0035)=0.6$$

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