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 answer:

n =  41

Step-by-step explanation:

$$\begin{gathered} p_1={\displaystyle \frac{1}{365}}\doteq 0.0027 \\ {q}_2=1-{\displaystyle \frac{364}{365}}={\displaystyle \frac{1}{365}}\doteq 0.0027 \\ {q}_3=1-(1-q_2)\cdot {\displaystyle \frac{363}{365}}=1-(1-0.0027)\cdot {\displaystyle \frac{363}{365}}\doteq 0.0082 \\ {q}_4=1-(1-q_3)\cdot {\displaystyle \frac{362}{365}}=1-(1-0.0082)\cdot {\displaystyle \frac{362}{365}}\doteq 0.0164 \\ {q}_5=1-(1-q_4)\cdot {\displaystyle \frac{365-4}{365}}=1-(1-0.0164)\cdot {\displaystyle \frac{365-4}{365}}\doteq 0.0271 \\ {q}_6=1-(1-q_5)\cdot {\displaystyle \frac{365-5}{365}}=1-(1-0.0271)\cdot {\displaystyle \frac{365-5}{365}}\doteq 0.0405 \\ {q}_7=1-(1-q_6)\cdot {\displaystyle \frac{365-6}{365}}=1-(1-0.0405)\cdot {\displaystyle \frac{365-6}{365}}\doteq 0.0562 \\ {q}_8=1-(1-q_7)\cdot {\displaystyle \frac{365-7}{365}}=1-(1-0.0562)\cdot {\displaystyle \frac{365-7}{365}}\doteq 0.0743 \\ {q}_9=1-(1-q_8)\cdot {\displaystyle \frac{365-8}{365}}=1-(1-0.0743)\cdot {\displaystyle \frac{365-8}{365}}\doteq 0.0946 \\ {q}_{10}=1-(1-q_9)\cdot {\displaystyle \frac{365-9}{365}}=1-(1-0.0946)\cdot {\displaystyle \frac{365-9}{365}}\doteq 0.1169 \\ {q}_{11}=1-(1-q_{10})\cdot {\displaystyle \frac{365-10}{365}}=1-(1-0.1169)\cdot {\displaystyle \frac{365-10}{365}}\doteq 0.1411 \\ {q}_{12}=1-(1-q_{11})\cdot {\displaystyle \frac{365-11}{365}}=1-(1-0.1411)\cdot {\displaystyle \frac{365-11}{365}}\doteq 0.167 \\ {q}_{13}=1-(1-q_{12})\cdot {\displaystyle \frac{365-12}{365}}=1-(1-0.167)\cdot {\displaystyle \frac{365-12}{365}}\doteq 0.1944 \\ {q}_{14}=1-(1-q_{13})\cdot {\displaystyle \frac{365-13}{365}}=1-(1-0.1944)\cdot {\displaystyle \frac{365-13}{365}}\doteq 0.2231 \\ {q}_{15}=1-(1-q_{14})\cdot {\displaystyle \frac{365-14}{365}}=1-(1-0.2231)\cdot {\displaystyle \frac{365-14}{365}}\doteq 0.2529 \\ {q}_{16}=1-(1-q_{15})\cdot {\displaystyle \frac{365-15}{365}}=1-(1-0.2529)\cdot {\displaystyle \frac{365-15}{365}}\doteq 0.2836 \\ {q}_{17}=1-(1-q_{16})\cdot {\displaystyle \frac{365-16}{365}}=1-(1-0.2836)\cdot {\displaystyle \frac{365-16}{365}}\doteq 0.315 \\ {q}_{18}=1-(1-q_{17})\cdot {\displaystyle \frac{365-17}{365}}=1-(1-0.315)\cdot {\displaystyle \frac{365-17}{365}}\doteq 0.3469 \\ {q}_{19}=1-(1-q_{18})\cdot {\displaystyle \frac{365-18}{365}}=1-(1-0.3469)\cdot {\displaystyle \frac{365-18}{365}}\doteq 0.3791 \\ {q}_{20}=1-(1-q_{19})\cdot {\displaystyle \frac{365-19}{365}}=1-(1-0.3791)\cdot {\displaystyle \frac{365-19}{365}}\doteq 0.4114 \\ {q}_{21}=1-(1-q_{20})\cdot {\displaystyle \frac{365-20}{365}}=1-(1-0.4114)\cdot {\displaystyle \frac{365-20}{365}}\doteq 0.4437 \\ {q}_{22}=1-(1-q_{21})\cdot {\displaystyle \frac{365-21}{365}}=1-(1-0.4437)\cdot {\displaystyle \frac{365-21}{365}}\doteq 0.4757 \\ {q}_{23}=1-(1-q_{22})\cdot {\displaystyle \frac{365-22}{365}}=1-(1-0.4757)\cdot {\displaystyle \frac{365-22}{365}}\doteq 0.5073 \\ {q}_{24}=1-(1-q_{23})\cdot {\displaystyle \frac{365-23}{365}}=1-(1-0.5073)\cdot {\displaystyle \frac{365-23}{365}}\doteq 0.5383 \\ {q}_{25}=1-(1-q_{24})\cdot {\displaystyle \frac{365-24}{365}}=1-(1-0.5383)\cdot {\displaystyle \frac{365-24}{365}}\doteq 0.5687 \\ {q}_{26}=1-(1-q_{25})\cdot {\displaystyle \frac{365-25}{365}}=1-(1-0.5687)\cdot {\displaystyle \frac{365-25}{365}}\doteq 0.5982 \\ {q}_{27}=1-(1-q_{26})\cdot {\displaystyle \frac{365-26}{365}}=1-(1-0.5982)\cdot {\displaystyle \frac{365-26}{365}}\doteq 0.6269 \\ {q}_{28}=1-(1-q_{27})\cdot {\displaystyle \frac{365-27}{365}}=1-(1-0.6269)\cdot {\displaystyle \frac{365-27}{365}}\doteq 0.6545 \\ {q}_{29}=1-(1-q_{28})\cdot {\displaystyle \frac{365-28}{365}}=1-(1-0.6545)\cdot {\displaystyle \frac{365-28}{365}}\doteq 0.681 \\ {q}_{30}=1-(1-q_{29})\cdot {\displaystyle \frac{365-29}{365}}=1-(1-0.681)\cdot {\displaystyle \frac{365-29}{365}}\doteq 0.7063 \\ {q}_{31}=1-(1-q_{30})\cdot {\displaystyle \frac{365-30}{365}}=1-(1-0.7063)\cdot {\displaystyle \frac{365-30}{365}}\doteq 0.7305 \\ {q}_{32}=1-(1-q_{31})\cdot {\displaystyle \frac{365-31}{365}}=1-(1-0.7305)\cdot {\displaystyle \frac{365-31}{365}}\doteq 0.7533 \\ {q}_{33}=1-(1-q_{32})\cdot {\displaystyle \frac{365-32}{365}}=1-(1-0.7533)\cdot {\displaystyle \frac{365-32}{365}}\doteq 0.775 \\ {q}_{34}=1-(1-q_{33})\cdot {\displaystyle \frac{365-33}{365}}=1-(1-0.775)\cdot {\displaystyle \frac{365-33}{365}}\doteq 0.7953 \\ {q}_{35}=1-(1-q_{34})\cdot {\displaystyle \frac{365-34}{365}}=1-(1-0.7953)\cdot {\displaystyle \frac{365-34}{365}}\doteq 0.8144 \\ {q}_{36}=1-(1-q_{35})\cdot {\displaystyle \frac{365-35}{365}}=1-(1-0.8144)\cdot {\displaystyle \frac{365-35}{365}}\doteq 0.8322 \\ {q}_{37}=1-(1-q_{36})\cdot {\displaystyle \frac{365-36}{365}}=1-(1-0.8322)\cdot {\displaystyle \frac{365-36}{365}}\doteq 0.8487 \\ {q}_{38}=1-(1-q_{37})\cdot {\displaystyle \frac{365-37}{365}}=1-(1-0.8487)\cdot {\displaystyle \frac{365-37}{365}}\doteq 0.8641 \\ {q}_{39}=1-(1-q_{38})\cdot {\displaystyle \frac{365-38}{365}}=1-(1-0.8641)\cdot {\displaystyle \frac{365-38}{365}}\doteq 0.8782 \\ {q}_{40}=1-(1-q_{39})\cdot {\displaystyle \frac{365-39}{365}}=1-(1-0.8782)\cdot {\displaystyle \frac{365-39}{365}}\doteq 0.8912 \\ {q}_{41}=1-(1-q_{40})\cdot {\displaystyle \frac{365-40}{365}}=1-(1-0.8912)\cdot {\displaystyle \frac{365-40}{365}}\doteq 0.9032 \\ {q}_{42}=1-(1-q_{41})\cdot {\displaystyle \frac{365-41}{365}}=1-(1-0.9032)\cdot {\displaystyle \frac{365-41}{365}}\doteq 0.914 \\ n=41 \end{gathered}$$

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