Question

There are 5% defective items in large bulk of items. Find the probability that a sample of 5 items will include not more than 2 defective items.

Solution

Here n = 5
$straight p space equals space 5 over 100 space equals space 1 over 20 comma space space straight q space equals space 1 space minus space straight p space equals space 1 space minus space 1 over 20 space equals space 19 over 20$
P(not more than 2 defective items) = P(0) + P(1) + P(2)
$equals space straight C presuperscript 5 subscript 0 space open parentheses 19 over 20 close parentheses to the power of 5 space plus space straight C presuperscript 5 subscript 1 space open parentheses 19 over 20 close parentheses to the power of 4 space open parentheses 1 over 20 close parentheses space plus space straight C presuperscript 5 subscript 2 space open parentheses 19 over 20 close parentheses cubed space open parentheses 1 over 20 close parentheses squared equals space open parentheses 19 over 20 close parentheses cubed space open square brackets straight C presuperscript 5 subscript 0 cross times open parentheses 19 over 20 close parentheses squared space plus space straight C presuperscript 5 subscript 1 space cross times space open parentheses 19 over 20 close parentheses space cross times space open parentheses 1 over 20 close parentheses space plus space straight C presuperscript 5 subscript 2 cross times open parentheses 1 over 20 close parentheses squared close square brackets equals space open parentheses 19 over 20 close parentheses cubed space open square brackets 1 space cross times space 361 over 400 plus space 5 over 1 cross times 19 over 400 plus fraction numerator 5 space cross times space 4 over denominator 1 space cross times 2 end fraction cross times 1 over 400 close square brackets space equals open parentheses 19 over 20 close parentheses cubed space open parentheses fraction numerator 361 plus 45 plus 10 over denominator 400 end fraction close parentheses space equals open parentheses 19 over 20 close parentheses cubed space cross times space 466 over 400 space equals open parentheses 19 over 20 close parentheses cubed space cross times 233 over 200$

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Probability

There are 5% defective items in large bulk of items. Find the probability that a sample of 5 items will include not more than 2 defective items.

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