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%?

Final 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

Try another example


Did you find an error or inaccuracy? Feel free to send us. Thank you!







Tips for related online calculators
Our percentage calculator will help you quickly and easily solve a variety of common percentage-related problems.
Do you want to convert time units like minutes to seconds?
Would you like to compute the count of combinations?

You need to know the following knowledge to solve this word math problem:

combinatoricsbasic operations and conceptsUnits of physical quantitiesthemes, topicsGrade of the word problem

Related math problems and questions: