Birthday paradox

How large is the group of people 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:

p1​=3651​≐0.0027  q2​=1−365364​=3651⋅ 365​−365364​=365365​−365364​=365365−364​=3651​≐0.0027 q3​=1−(1−q2​)⋅ 365363​=1−(1−3651​)⋅ 365363​≐0.0082 q4​=1−(1−q3​)⋅ 365362​=1−(1−0.0082)⋅ 365362​≐0.0164  q5​=1−(1−q4​)⋅ 365365−4​=1−(1−0.0164)⋅ 365365−4​≐0.0271 q6​=1−(1−q5​)⋅ 365365−5​=1−(1−0.0271)⋅ 365365−5​≐0.0405 q7​=1−(1−q6​)⋅ 365365−6​=1−(1−0.0405)⋅ 365365−6​≐0.0562 q8​=1−(1−q7​)⋅ 365365−7​=1−(1−0.0562)⋅ 365365−7​≐0.0743 q9​=1−(1−q8​)⋅ 365365−8​=1−(1−0.0743)⋅ 365365−8​≐0.0946 q10​=1−(1−q9​)⋅ 365365−9​=1−(1−0.0946)⋅ 365365−9​≐0.1169 q11​=1−(1−q10​)⋅ 365365−10​=1−(1−0.1169)⋅ 365365−10​≐0.1411 q12​=1−(1−q11​)⋅ 365365−11​=1−(1−0.1411)⋅ 365365−11​≐0.167 q13​=1−(1−q12​)⋅ 365365−12​=1−(1−0.167)⋅ 365365−12​≐0.1944 q14​=1−(1−q13​)⋅ 365365−13​=1−(1−0.1944)⋅ 365365−13​≐0.2231 q15​=1−(1−q14​)⋅ 365365−14​=1−(1−0.2231)⋅ 365365−14​≐0.2529 q16​=1−(1−q15​)⋅ 365365−15​=1−(1−0.2529)⋅ 365365−15​≐0.2836 q17​=1−(1−q16​)⋅ 365365−16​=1−(1−0.2836)⋅ 365365−16​≐0.315 q18​=1−(1−q17​)⋅ 365365−17​=1−(1−0.315)⋅ 365365−17​≐0.3469 q19​=1−(1−q18​)⋅ 365365−18​=1−(1−0.3469)⋅ 365365−18​≐0.3791 q20​=1−(1−q19​)⋅ 365365−19​=1−(1−0.3791)⋅ 365365−19​≐0.4114 q21​=1−(1−q20​)⋅ 365365−20​=1−(1−0.4114)⋅ 365365−20​≐0.4437 q22​=1−(1−q21​)⋅ 365365−21​=1−(1−0.4437)⋅ 365365−21​≐0.4757 q23​=1−(1−q22​)⋅ 365365−22​=1−(1−0.4757)⋅ 365365−22​≐0.5073 q24​=1−(1−q23​)⋅ 365365−23​=1−(1−0.5073)⋅ 365365−23​≐0.5383 q25​=1−(1−q24​)⋅ 365365−24​=1−(1−0.5383)⋅ 365365−24​≐0.5687 q26​=1−(1−q25​)⋅ 365365−25​=1−(1−0.5687)⋅ 365365−25​≐0.5982 q27​=1−(1−q26​)⋅ 365365−26​=1−(1−0.5982)⋅ 365365−26​≐0.6269 q28​=1−(1−q27​)⋅ 365365−27​=1−(1−0.6269)⋅ 365365−27​≐0.6545 q29​=1−(1−q28​)⋅ 365365−28​=1−(1−0.6545)⋅ 365365−28​≐0.681 q30​=1−(1−q29​)⋅ 365365−29​=1−(1−0.681)⋅ 365365−29​≐0.7063 q31​=1−(1−q30​)⋅ 365365−30​=1−(1−0.7063)⋅ 365365−30​≐0.7305 q32​=1−(1−q31​)⋅ 365365−31​=1−(1−0.7305)⋅ 365365−31​≐0.7533 q33​=1−(1−q32​)⋅ 365365−32​=1−(1−0.7533)⋅ 365365−32​≐0.775 q34​=1−(1−q33​)⋅ 365365−33​=1−(1−0.775)⋅ 365365−33​≐0.7953 q35​=1−(1−q34​)⋅ 365365−34​=1−(1−0.7953)⋅ 365365−34​≐0.8144 q36​=1−(1−q35​)⋅ 365365−35​=1−(1−0.8144)⋅ 365365−35​≐0.8322 q37​=1−(1−q36​)⋅ 365365−36​=1−(1−0.8322)⋅ 365365−36​≐0.8487 q38​=1−(1−q37​)⋅ 365365−37​=1−(1−0.8487)⋅ 365365−37​≐0.8641 q39​=1−(1−q38​)⋅ 365365−38​=1−(1−0.8641)⋅ 365365−38​≐0.8782 q40​=1−(1−q39​)⋅ 365365−39​=1−(1−0.8782)⋅ 365365−39​≐0.8912 q41​=1−(1−q40​)⋅ 365365−40​=1−(1−0.8912)⋅ 365365−40​≐0.9032 q42​=1−(1−q41​)⋅ 365365−41​=1−(1−0.9032)⋅ 365365−41​≐0.914 n=41

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