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:

C_{0} (800) = (\binom{800}{0}) = \frac{800!}{0! (800 - 0)!} = \frac{1}{1} = 1 C_{1} (800) = (\binom{800}{1}) = \frac{800!}{1! (800 - 1)!} = \frac{800}{1} = 800 C_{2} (800) = (\binom{800}{2}) = \frac{800!}{2! (800 - 2)!} = \frac{800 \cdot 799}{2 \cdot 1} = 319600 C_{3} (800) = (\binom{800}{3}) = \frac{800!}{3! (800 - 3)!} = \frac{800 \cdot 799 \cdot 798}{3 \cdot 2 \cdot 1} = 85013600 C_{4} (800) = (\binom{800}{4}) = \frac{800!}{4! (800 - 4)!} = \frac{800 \cdot 799 \cdot 798 \cdot 797}{4 \cdot 3 \cdot 2 \cdot 1} = 16938959800 C_{5} (800) = (\binom{800}{5}) = \frac{800!}{5! (800 - 5)!} = \frac{800 \cdot 799 \cdot 798 \cdot 797 \cdot 796}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = 2696682400160 C_{6} (800) = (\binom{800}{6}) = \frac{800!}{6! (800 - 6)!} = \frac{800 \cdot 799 \cdot 798 \cdot 797 \cdot 796 \cdot 795}{6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = 357310418021200 C_{7} (800) = (\binom{800}{7}) = \frac{800!}{7! (800 - 7)!} = \frac{800 \cdot 799 \cdot 798 \cdot 797 \cdot 796 \cdot 795 \cdot 794}{7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = 40529210272690400 C_{8} (800) = (\binom{800}{8}) = \frac{800!}{8! (800 - 8)!} = \frac{800 \cdot 799 \cdot 798 \cdot 797 \cdot 796 \cdot 795 \cdot 794 \cdot 793}{8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = 4017457968280435900 C_{9} (800) = (\binom{800}{9}) = \frac{800!}{9! (800 - 9)!} \approx 3.535 \times 1 0^{20} = 353536301208678359200 C_{10} (800) = (\binom{800}{10}) = \frac{800!}{10! (800 - 10)!} \approx 2.796 \times 1 0^{22} = 27964721425606458212720 C_{11} (800) = (\binom{800}{11}) = \frac{800!}{11! (800 - 11)!} \approx 2.008 \times 1 0^{24} = 2008375447839009271640800 C_{12} (800) = (\binom{800}{12}) = \frac{800!}{12! (800 - 12)!} \approx 1.320 \times 1 0^{26} = 132050685695414859610382600 C_{13} (800) = (\binom{800}{13}) = \frac{800!}{13! (800 - 13)!} \approx 8.004 \times 1 0^{27} = 8004303102152839182537037600 C_{14} (800) = (\binom{800}{14}) = \frac{800!}{14! (800 - 14)!} \approx 4.499 \times 1 0^{29} = 449956181528163174046903470800 C_{15} (800) = (\binom{800}{15}) = \frac{800!}{15! (800 - 15)!} \approx 2.357 \times 1 0^{31} = 23577703912075750320057741869920 C_{16} (800) = (\binom{800}{16}) = \frac{800!}{16! (800 - 16)!} \approx 1.156 \times 1 0^{33} = 1156781098186216500077832960492950 C_{17} (800) = (\binom{800}{17}) = \frac{800!}{17! (800 - 17)!} \approx 5.334 \times 1 0^{34} = 53348022410470219768295355354498400 C_{18} (800) = (\binom{800}{18}) = \frac{800!}{18! (800 - 18)!} \approx 2.320 \times 1 0^{36} = 2320638974855454559920847957920680400 C_{19} (800) = (\binom{800}{19}) = \frac{800!}{19! (800 - 19)!} \approx 9.551 \times 1 0^{37} = 95512614649313971887268584373366951200 q = 2 % = \frac{2}{100} = 0.02 n = 800 d = 20 p_{0} = (\binom{n}{0}) \cdot q^{0} \cdot (1 - q)^{n - 0} = 1 \cdot 0.0 2^{0} \cdot (1 - 0.02)^{800 - 0} ≐ 9.5689 \cdot 1 0^{- 8} p_{1} = (\binom{n}{1}) \cdot q^{1} \cdot (1 - q)^{n - 1} = 800 \cdot 0.0 2^{1} \cdot (1 - 0.02)^{800 - 1} ≐ 1.5623 \cdot 1 0^{- 6} p_{2} = (\binom{n}{2}) \cdot q^{2} \cdot (1 - q)^{n - 2} = 319600 \cdot 0.0 2^{2} \cdot (1 - 0.02)^{800 - 2} ≐ 1.2737 \cdot 1 0^{- 5} p_{3} = (\binom{n}{3}) \cdot q^{3} \cdot (1 - q)^{n - 3} = 85013600 \cdot 0.0 2^{3} \cdot (1 - 0.02)^{800 - 3} ≐ 0.0001 p_{4} = (\binom{n}{4}) \cdot q^{4} \cdot (1 - q)^{n - 4} = 16938959800 \cdot 0.0 2^{4} \cdot (1 - 0.02)^{800 - 4} ≐ 0.0003 p_{5} = (\binom{n}{5}) \cdot q^{5} \cdot (1 - q)^{n - 5} = 2696682400160 \cdot 0.0 2^{5} \cdot (1 - 0.02)^{800 - 5} ≐ 0.0009 p_{6} = (\binom{n}{6}) \cdot q^{6} \cdot (1 - q)^{n - 6} = 357310418021200 \cdot 0.0 2^{6} \cdot (1 - 0.02)^{800 - 6} ≐ 0.0025 p_{7} = (\binom{n}{7}) \cdot q^{7} \cdot (1 - q)^{n - 7} = 40529210272690400 \cdot 0.0 2^{7} \cdot (1 - 0.02)^{800 - 7} ≐ 0.0057 p_{8} = (\binom{n}{8}) \cdot q^{8} \cdot (1 - q)^{n - 8} = 4017457968280435900 \cdot 0.0 2^{8} \cdot (1 - 0.02)^{800 - 8} ≐ 0.0116 p_{9} = (\binom{n}{9}) \cdot q^{9} \cdot (1 - q)^{n - 9} = 353536301208678359200 \cdot 0.0 2^{9} \cdot (1 - 0.02)^{800 - 9} ≐ 0.0208 p_{10} = (\binom{n}{10}) \cdot q^{10} \cdot (1 - q)^{n - 10} = 27964721425606458212720 \cdot 0.0 2^{10} \cdot (1 - 0.02)^{800 - 10} ≐ 0.0335 p_{11} = (\binom{n}{11}) \cdot q^{11} \cdot (1 - q)^{n - 11} = 2008375447839009271640800 \cdot 0.0 2^{11} \cdot (1 - 0.02)^{800 - 11} ≐ 0.0492 p_{12} = (\binom{n}{12}) \cdot q^{12} \cdot (1 - q)^{n - 12} = 132050685695414859610382600 \cdot 0.0 2^{12} \cdot (1 - 0.02)^{800 - 12} ≐ 0.066 p_{13} = (\binom{n}{13}) \cdot q^{13} \cdot (1 - q)^{n - 13} = 8004303102152839182537037600 \cdot 0.0 2^{13} \cdot (1 - 0.02)^{800 - 13} ≐ 0.0816 p_{14} = (\binom{n}{14}) \cdot q^{14} \cdot (1 - q)^{n - 14} = 449956181528163174046903470800 \cdot 0.0 2^{14} \cdot (1 - 0.02)^{800 - 14} ≐ 0.0936 p_{15} = (\binom{n}{15}) \cdot q^{15} \cdot (1 - q)^{n - 15} = 23577703912075750320057741869920 \cdot 0.0 2^{15} \cdot (1 - 0.02)^{800 - 15} ≐ 0.1001 p_{16} = (\binom{n}{16}) \cdot q^{16} \cdot (1 - q)^{n - 16} = 1156781098186216500077832960492950 \cdot 0.0 2^{16} \cdot (1 - 0.02)^{800 - 16} ≐ 0.1002 p_{17} = (\binom{n}{17}) \cdot q^{17} \cdot (1 - q)^{n - 17} = 53348022410470219768295355354498400 \cdot 0.0 2^{17} \cdot (1 - 0.02)^{800 - 17} ≐ 0.0943 p_{18} = (\binom{n}{18}) \cdot q^{18} \cdot (1 - q)^{n - 18} = 2320638974855454559920847957920680400 \cdot 0.0 2^{18} \cdot (1 - 0.02)^{800 - 18} ≐ 0.0837 p_{19} = (\binom{n}{19}) \cdot q^{19} \cdot (1 - q)^{n - 19} = 95512614649313971887268584373366951200 \cdot 0.0 2^{19} \cdot (1 - 0.02)^{800 - 19} ≐ 0.0703 p = p_{0} + p_{1} + p_{2} + p_{3} + p_{4} + p_{5} + p_{6} + p_{7} + p_{8} + p_{9} + p_{10} + p_{11} + p_{12} + p_{13} + p_{14} + p_{15} + p_{16} + p_{17} + p_{18} + p_{19} = 9.5689 \cdot 1 0^{- 8} + 1.5623 \cdot 1 0^{- 6} + 1.2737 \cdot 1 0^{- 5} + 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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