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%?

Result

n =  41

Solution:

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

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