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Probability

Question
CBSEENMA12033855
Wired Faculty App

There are 8 % defective items in a large bulk of items. Find the probability that a sample of 10 items will include not more than 3 defective items.

Solution

Here n = 10
                 straight p space equals 8 over 100 space equals 2 over 25 comma space space space straight q space equals space 1 space minus space straight p space equals space 1 minus 2 over 25 space equals space 23 over 25
P(not more than 3 defective items) = P(0) + P(1) + P(2) + P(3)
                        equals space straight C presuperscript 10 subscript 0 open parentheses 23 over 25 close parentheses to the power of 10 space plus space straight C presuperscript 10 subscript 1 space open parentheses 2 over 25 close parentheses space open parentheses 23 over 25 close parentheses to the power of 9 space plus space straight C presuperscript 10 subscript 2 space open parentheses 2 over 25 close parentheses squared space open parentheses 23 over 25 close parentheses to the power of 8 space plus space straight C presuperscript 10 subscript 3 space open parentheses 2 over 25 close parentheses cubed space open parentheses 23 over 25 close parentheses to the power of 7 equals space 1 space cross times space open parentheses 23 over 25 close parentheses to the power of 10 space plus space 10 over 1 cross times 2 over 25 cross times space open parentheses 23 over 25 close parentheses to the power of 9 space plus space fraction numerator 10 space cross times 9 over denominator 1 cross times 2 end fraction cross times space open parentheses 2 over 25 close parentheses squared space cross times open parentheses 23 over 25 close parentheses to the power of 8 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space plus space fraction numerator 10 cross times 9 cross times 8 over denominator 1 cross times 2 cross times 3 end fraction open parentheses 2 over 25 close parentheses cubed space space open parentheses 23 over 25 close parentheses to the power of 7 equals space open parentheses 23 over 25 close parentheses to the power of 7 space open square brackets open parentheses 23 over 25 close parentheses cubed plus 4 over 5 cross times open parentheses 23 over 25 close parentheses squared space plus space 45 space cross times space open parentheses 2 over 25 close parentheses squared space cross times 23 over 25 plus 120 space cross times space open parentheses 2 over 25 close parentheses cubed close square brackets equals space fraction numerator left parenthesis 23 right parenthesis to the power of 7 over denominator left parenthesis 25 right parenthesis to the power of 10 end fraction space open square brackets left parenthesis 23 right parenthesis squared plus space 20 space cross times space left parenthesis 23 right parenthesis squared space plus space 45 space cross times space 4 space cross times 23 space plus 120 space cross times 8 close square brackets equals space fraction numerator left parenthesis 23 right parenthesis to the power of 7 over denominator left parenthesis 25 right parenthesis to the power of 10 end fraction left square bracket 12167 space plus space 10580 space plus space 4140 space plus space 960 right square bracket space equals space 27847 space open square brackets fraction numerator left parenthesis 23 right parenthesis to the power of 7 over denominator left parenthesis 25 right parenthesis to the power of 10 end fraction close square brackets

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