Birthday paradox

How large must the group of people be so that the probability that two people have a birthday on the same day of the year is greater than 90%?

Correct result:

n = 41

Solution:

p_{1} = \frac{1}{365} ≐ 0.0027 q_{2} = 1 - \frac{364}{365} = \frac{1}{365} ≐ 0.0027 q_{3} = 1 - (1 - q_{2}) \cdot \frac{363}{365} = 1 - (1 - 0.0027) \cdot \frac{363}{365} ≐ 0.0082 q_{4} = 1 - (1 - q_{3}) \cdot \frac{362}{365} = 1 - (1 - 0.0082) \cdot \frac{362}{365} ≐ 0.0164 q_{5} = 1 - (1 - q_{4}) \cdot \frac{365 - 4}{365} = 1 - (1 - 0.0164) \cdot \frac{365 - 4}{365} ≐ 0.0271 q_{6} = 1 - (1 - q_{5}) \cdot \frac{365 - 5}{365} = 1 - (1 - 0.0271) \cdot \frac{365 - 5}{365} ≐ 0.0405 q_{7} = 1 - (1 - q_{6}) \cdot \frac{365 - 6}{365} = 1 - (1 - 0.0405) \cdot \frac{365 - 6}{365} ≐ 0.0562 q_{8} = 1 - (1 - q_{7}) \cdot \frac{365 - 7}{365} = 1 - (1 - 0.0562) \cdot \frac{365 - 7}{365} ≐ 0.0743 q_{9} = 1 - (1 - q_{8}) \cdot \frac{365 - 8}{365} = 1 - (1 - 0.0743) \cdot \frac{365 - 8}{365} ≐ 0.0946 q_{10} = 1 - (1 - q_{9}) \cdot \frac{365 - 9}{365} = 1 - (1 - 0.0946) \cdot \frac{365 - 9}{365} ≐ 0.1169 q_{11} = 1 - (1 - q_{10}) \cdot \frac{365 - 10}{365} = 1 - (1 - 0.1169) \cdot \frac{365 - 10}{365} ≐ 0.1411 q_{12} = 1 - (1 - q_{11}) \cdot \frac{365 - 11}{365} = 1 - (1 - 0.1411) \cdot \frac{365 - 11}{365} ≐ 0.167 q_{13} = 1 - (1 - q_{12}) \cdot \frac{365 - 12}{365} = 1 - (1 - 0.167) \cdot \frac{365 - 12}{365} ≐ 0.1944 q_{14} = 1 - (1 - q_{13}) \cdot \frac{365 - 13}{365} = 1 - (1 - 0.1944) \cdot \frac{365 - 13}{365} ≐ 0.2231 q_{15} = 1 - (1 - q_{14}) \cdot \frac{365 - 14}{365} = 1 - (1 - 0.2231) \cdot \frac{365 - 14}{365} ≐ 0.2529 q_{16} = 1 - (1 - q_{15}) \cdot \frac{365 - 15}{365} = 1 - (1 - 0.2529) \cdot \frac{365 - 15}{365} ≐ 0.2836 q_{17} = 1 - (1 - q_{16}) \cdot \frac{365 - 16}{365} = 1 - (1 - 0.2836) \cdot \frac{365 - 16}{365} ≐ 0.315 q_{18} = 1 - (1 - q_{17}) \cdot \frac{365 - 17}{365} = 1 - (1 - 0.315) \cdot \frac{365 - 17}{365} ≐ 0.3469 q_{19} = 1 - (1 - q_{18}) \cdot \frac{365 - 18}{365} = 1 - (1 - 0.3469) \cdot \frac{365 - 18}{365} ≐ 0.3791 q_{20} = 1 - (1 - q_{19}) \cdot \frac{365 - 19}{365} = 1 - (1 - 0.3791) \cdot \frac{365 - 19}{365} ≐ 0.4114 q_{21} = 1 - (1 - q_{20}) \cdot \frac{365 - 20}{365} = 1 - (1 - 0.4114) \cdot \frac{365 - 20}{365} ≐ 0.4437 q_{22} = 1 - (1 - q_{21}) \cdot \frac{365 - 21}{365} = 1 - (1 - 0.4437) \cdot \frac{365 - 21}{365} ≐ 0.4757 q_{23} = 1 - (1 - q_{22}) \cdot \frac{365 - 22}{365} = 1 - (1 - 0.4757) \cdot \frac{365 - 22}{365} ≐ 0.5073 q_{24} = 1 - (1 - q_{23}) \cdot \frac{365 - 23}{365} = 1 - (1 - 0.5073) \cdot \frac{365 - 23}{365} ≐ 0.5383 q_{25} = 1 - (1 - q_{24}) \cdot \frac{365 - 24}{365} = 1 - (1 - 0.5383) \cdot \frac{365 - 24}{365} ≐ 0.5687 q_{26} = 1 - (1 - q_{25}) \cdot \frac{365 - 25}{365} = 1 - (1 - 0.5687) \cdot \frac{365 - 25}{365} ≐ 0.5982 q_{27} = 1 - (1 - q_{26}) \cdot \frac{365 - 26}{365} = 1 - (1 - 0.5982) \cdot \frac{365 - 26}{365} ≐ 0.6269 q_{28} = 1 - (1 - q_{27}) \cdot \frac{365 - 27}{365} = 1 - (1 - 0.6269) \cdot \frac{365 - 27}{365} ≐ 0.6545 q_{29} = 1 - (1 - q_{28}) \cdot \frac{365 - 28}{365} = 1 - (1 - 0.6545) \cdot \frac{365 - 28}{365} ≐ 0.681 q_{30} = 1 - (1 - q_{29}) \cdot \frac{365 - 29}{365} = 1 - (1 - 0.681) \cdot \frac{365 - 29}{365} ≐ 0.7063 q_{31} = 1 - (1 - q_{30}) \cdot \frac{365 - 30}{365} = 1 - (1 - 0.7063) \cdot \frac{365 - 30}{365} ≐ 0.7305 q_{32} = 1 - (1 - q_{31}) \cdot \frac{365 - 31}{365} = 1 - (1 - 0.7305) \cdot \frac{365 - 31}{365} ≐ 0.7533 q_{33} = 1 - (1 - q_{32}) \cdot \frac{365 - 32}{365} = 1 - (1 - 0.7533) \cdot \frac{365 - 32}{365} ≐ 0.775 q_{34} = 1 - (1 - q_{33}) \cdot \frac{365 - 33}{365} = 1 - (1 - 0.775) \cdot \frac{365 - 33}{365} ≐ 0.7953 q_{35} = 1 - (1 - q_{34}) \cdot \frac{365 - 34}{365} = 1 - (1 - 0.7953) \cdot \frac{365 - 34}{365} ≐ 0.8144 q_{36} = 1 - (1 - q_{35}) \cdot \frac{365 - 35}{365} = 1 - (1 - 0.8144) \cdot \frac{365 - 35}{365} ≐ 0.8322 q_{37} = 1 - (1 - q_{36}) \cdot \frac{365 - 36}{365} = 1 - (1 - 0.8322) \cdot \frac{365 - 36}{365} ≐ 0.8487 q_{38} = 1 - (1 - q_{37}) \cdot \frac{365 - 37}{365} = 1 - (1 - 0.8487) \cdot \frac{365 - 37}{365} ≐ 0.8641 q_{39} = 1 - (1 - q_{38}) \cdot \frac{365 - 38}{365} = 1 - (1 - 0.8641) \cdot \frac{365 - 38}{365} ≐ 0.8782 q_{40} = 1 - (1 - q_{39}) \cdot \frac{365 - 39}{365} = 1 - (1 - 0.8782) \cdot \frac{365 - 39}{365} ≐ 0.8912 q_{41} = 1 - (1 - q_{40}) \cdot \frac{365 - 40}{365} = 1 - (1 - 0.8912) \cdot \frac{365 - 40}{365} ≐ 0.9032 q_{42} = 1 - (1 - q_{41}) \cdot \frac{365 - 41}{365} = 1 - (1 - 0.9032) \cdot \frac{365 - 41}{365} ≐ 0.914 n = 41

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