Bernoulli distribution

The production of solar cells produces 2% of defective cells. Assume the cells are independent and that a lot contains 800 cells. Approximate the probability that less than 20 cells are defective. (Answer to the nearest 3 decimals).

Correct result:

p =  0.814

Solution:

C0(800)=(8000)=800!0!(8000)!=11=1 C1(800)=(8001)=800!1!(8001)!=8001=800 C2(800)=(8002)=800!2!(8002)!=80079921=319600 C3(800)=(8003)=800!3!(8003)!=800799798321=85013600 C4(800)=(8004)=800!4!(8004)!=8007997987974321=16938959800 C5(800)=(8005)=800!5!(8005)!=80079979879779654321=2696682400160 C6(800)=(8006)=800!6!(8006)!=800799798797796795654321=357310418021200 C7(800)=(8007)=800!7!(8007)!=8007997987977967957947654321=40529210272690400 C8(800)=(8008)=800!8!(8008)!=80079979879779679579479387654321=4017457968280435900 C9(800)=(8009)=800!9!(8009)!3.535×1020=353536301208678359200 C10(800)=(80010)=800!10!(80010)!2.796×1022=27964721425606458212720 C11(800)=(80011)=800!11!(80011)!2.008×1024=2008375447839009271640800 C12(800)=(80012)=800!12!(80012)!1.320×1026=132050685695414859610382600 C13(800)=(80013)=800!13!(80013)!8.004×1027=8004303102152839182537037600 C14(800)=(80014)=800!14!(80014)!4.499×1029=449956181528163174046903470800 C15(800)=(80015)=800!15!(80015)!2.357×1031=23577703912075750320057741869920 C16(800)=(80016)=800!16!(80016)!1.156×1033=1156781098186216500077832960492950 C17(800)=(80017)=800!17!(80017)!5.334×1034=53348022410470219768295355354498400 C18(800)=(80018)=800!18!(80018)!2.320×1036=2320638974855454559920847957920680400 C19(800)=(80019)=800!19!(80019)!9.551×1037=95512614649313971887268584373366951200 q=2%=2100=0.02 n=800 d=20  p0=(n0) q0 (1q)n0=1 0.020 (10.02)80009.5689108 p1=(n1) q1 (1q)n1=800 0.021 (10.02)80011.5623106 p2=(n2) q2 (1q)n2=319600 0.022 (10.02)80021.2737105 p3=(n3) q3 (1q)n3=85013600 0.023 (10.02)80030.0001 p4=(n4) q4 (1q)n4=16938959800 0.024 (10.02)80040.0003 p5=(n5) q5 (1q)n5=2696682400160 0.025 (10.02)80050.0009 p6=(n6) q6 (1q)n6=357310418021200 0.026 (10.02)80060.0025 p7=(n7) q7 (1q)n7=40529210272690400 0.027 (10.02)80070.0057 p8=(n8) q8 (1q)n8=4017457968280435900 0.028 (10.02)80080.0116 p9=(n9) q9 (1q)n9=353536301208678359200 0.029 (10.02)80090.0208 p10=(n10) q10 (1q)n10=27964721425606458212720 0.0210 (10.02)800100.0335 p11=(n11) q11 (1q)n11=2008375447839009271640800 0.0211 (10.02)800110.0492 p12=(n12) q12 (1q)n12=132050685695414859610382600 0.0212 (10.02)800120.066 p13=(n13) q13 (1q)n13=8004303102152839182537037600 0.0213 (10.02)800130.0816 p14=(n14) q14 (1q)n14=449956181528163174046903470800 0.0214 (10.02)800140.0936 p15=(n15) q15 (1q)n15=23577703912075750320057741869920 0.0215 (10.02)800150.1001 p16=(n16) q16 (1q)n16=1156781098186216500077832960492950 0.0216 (10.02)800160.1002 p17=(n17) q17 (1q)n17=53348022410470219768295355354498400 0.0217 (10.02)800170.0943 p18=(n18) q18 (1q)n18=2320638974855454559920847957920680400 0.0218 (10.02)800180.0837 p19=(n19) q19 (1q)n19=95512614649313971887268584373366951200 0.0219 (10.02)800190.0703   p=p0+p1+p2+p3+p4+p5+p6+p7+p8+p9+p10+p11+p12+p13+p14+p15+p16+p17+p18+p19=9.5689108+1.5623106+1.2737105+0.0001+0.0003+0.0009+0.0025+0.0057+0.0116+0.0208+0.0335+0.0492+0.066+0.0816+0.0936+0.1001+0.1002+0.0943+0.0837+0.0703=0.814



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