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%?

Result

n =  41

Solution:

p1=1365≐0.0027  q2=1−364365=1365≐0.0027 q3=1−(1−q2)⋅ 363365=1−(1−0.0027)⋅ 363365≐0.0082 q4=1−(1−q3)⋅ 362365=1−(1−0.0082)⋅ 362365≐0.0164  q5=1−(1−q4)⋅ 365−4365=1−(1−0.0164)⋅ 365−4365≐0.0271 q6=1−(1−q5)⋅ 365−5365=1−(1−0.0271)⋅ 365−5365≐0.0405 q7=1−(1−q6)⋅ 365−6365=1−(1−0.0405)⋅ 365−6365≐0.0562 q8=1−(1−q7)⋅ 365−7365=1−(1−0.0562)⋅ 365−7365≐0.0743 q9=1−(1−q8)⋅ 365−8365=1−(1−0.0743)⋅ 365−8365≐0.0946 q10=1−(1−q9)⋅ 365−9365=1−(1−0.0946)⋅ 365−9365≐0.1169 q11=1−(1−q10)⋅ 365−10365=1−(1−0.1169)⋅ 365−10365≐0.1411 q12=1−(1−q11)⋅ 365−11365=1−(1−0.1411)⋅ 365−11365≐0.167 q13=1−(1−q12)⋅ 365−12365=1−(1−0.167)⋅ 365−12365≐0.1944 q14=1−(1−q13)⋅ 365−13365=1−(1−0.1944)⋅ 365−13365≐0.2231 q15=1−(1−q14)⋅ 365−14365=1−(1−0.2231)⋅ 365−14365≐0.2529 q16=1−(1−q15)⋅ 365−15365=1−(1−0.2529)⋅ 365−15365≐0.2836 q17=1−(1−q16)⋅ 365−16365=1−(1−0.2836)⋅ 365−16365≐0.315 q18=1−(1−q17)⋅ 365−17365=1−(1−0.315)⋅ 365−17365≐0.3469 q19=1−(1−q18)⋅ 365−18365=1−(1−0.3469)⋅ 365−18365≐0.3791 q20=1−(1−q19)⋅ 365−19365=1−(1−0.3791)⋅ 365−19365≐0.4114 q21=1−(1−q20)⋅ 365−20365=1−(1−0.4114)⋅ 365−20365≐0.4437 q22=1−(1−q21)⋅ 365−21365=1−(1−0.4437)⋅ 365−21365≐0.4757 q23=1−(1−q22)⋅ 365−22365=1−(1−0.4757)⋅ 365−22365≐0.5073 q24=1−(1−q23)⋅ 365−23365=1−(1−0.5073)⋅ 365−23365≐0.5383 q25=1−(1−q24)⋅ 365−24365=1−(1−0.5383)⋅ 365−24365≐0.5687 q26=1−(1−q25)⋅ 365−25365=1−(1−0.5687)⋅ 365−25365≐0.5982 q27=1−(1−q26)⋅ 365−26365=1−(1−0.5982)⋅ 365−26365≐0.6269 q28=1−(1−q27)⋅ 365−27365=1−(1−0.6269)⋅ 365−27365≐0.6545 q29=1−(1−q28)⋅ 365−28365=1−(1−0.6545)⋅ 365−28365≐0.681 q30=1−(1−q29)⋅ 365−29365=1−(1−0.681)⋅ 365−29365≐0.7063 q31=1−(1−q30)⋅ 365−30365=1−(1−0.7063)⋅ 365−30365≐0.7305 q32=1−(1−q31)⋅ 365−31365=1−(1−0.7305)⋅ 365−31365≐0.7533 q33=1−(1−q32)⋅ 365−32365=1−(1−0.7533)⋅ 365−32365≐0.775 q34=1−(1−q33)⋅ 365−33365=1−(1−0.775)⋅ 365−33365≐0.7953 q35=1−(1−q34)⋅ 365−34365=1−(1−0.7953)⋅ 365−34365≐0.8144 q36=1−(1−q35)⋅ 365−35365=1−(1−0.8144)⋅ 365−35365≐0.8322 q37=1−(1−q36)⋅ 365−36365=1−(1−0.8322)⋅ 365−36365≐0.8487 q38=1−(1−q37)⋅ 365−37365=1−(1−0.8487)⋅ 365−37365≐0.8641 q39=1−(1−q38)⋅ 365−38365=1−(1−0.8641)⋅ 365−38365≐0.8782 q40=1−(1−q39)⋅ 365−39365=1−(1−0.8782)⋅ 365−39365≐0.8912 q41=1−(1−q40)⋅ 365−40365=1−(1−0.8912)⋅ 365−40365≐0.9032 q42=1−(1−q41)⋅ 365−41365=1−(1−0.9032)⋅ 365−41365≐0.914 n=41p_{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

Try another example


Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!





Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):

Showing 0 comments:
1st comment
Be the first to comment!
avatar




Tips to related online calculators
Our percentage calculator will help you quickly calculate various typical tasks with percentages.
Do you want to convert time units like minutes to seconds?
Would you like to compute count of combinations?

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

Next similar math problems:

  1. Birth
    probability Let's assume that the probability of the birth of a boy and a girl in the family is the same. What is the probability that in a family with five children, the youngest and oldest child is a boy?
  2. Green - Red
    balls_2 We have 5 bags. Each consist one green and 2 red balls. From each we pull just one ball. What is the probability that we doesn't pull any green ball?
  3. Throw
    dices6_1 We throw 2 times with 2 dices. What is the probability that the first roll will fall more than sum of 9 and the second throw have sum 3 or does not have the sum 4?
  4. Balls
    spheres From the urn in which are 7 white balls and 17 red, gradually drag 3-times without replacement. What is the probability that pulls balls are in order: red red red?
  5. Cards
    cards_2 Suppose that are three cards in the hats. One is red on both sides, one of which is black on both sides, and a third one side red and the second black. We are pulled out of a hat randomly one card and we see that one side of it is red. What is the probab
  6. The test
    test The test contains four questions, and there are five different answers to each of them, of which only one is correct, the others are incorrect. What is the probability that a student who does not know the answer to any question will guess the right answer
  7. Two aces
    cards_7 From a 32 card box we randomly pick 1 card and then 2 more cards. What is the probability that last two drawn cards are aces?
  8. Bulbs
    bulb The probability that the bulb can operate 4000 hours is 0.3. What is the probability that exactly one of eight bulbs can operate 4000 hours?
  9. Win in raffle
    tombola_1 The raffle tickets were sold 200, 5 of which were winning. What is the probability that Peter, who bought one ticket will win?
  10. travel agency
    nevesta Small travel agency offers 5 different tours at honeymoon. What is the probability that the bride and groom choose the same tour (they choose independently)?
  11. Medicine
    lieky We test medicine on 6 patients. For all drug doesn't work. If the drug success rate of 20%, what is the probability that medicine does not work?
  12. A book
    books_32 A book contains 524 pages. If it is known that a person will select any one page between the pages numbered 125 and 384, find the probability of choosing the page numbered 252 or 253.
  13. Hearts
    hearts_cards 5 cards are chosen from a standard deck of 52 playing cards (13 hearts) with replacement. What is the probability of choosing 5 hearts in a row?
  14. Three shooters
    terc2_3 Three shooters shoot, each one time, on the same target. The first hit the target with a probability of 0.7; second with a probability of 0.8 and a third with a probability of 0.9. What is the probability to hit the target: a) just once b) at least once
  15. Shooters
    soldiers In army regiment are six shooters. The first shooter target hit with a probability of 49%, next with 75%, 41%, 20%, 34%, 63%. Calculate the probability of target hit when shooting all at once.
  16. Researchers
    house Researchers ask 200 families whether or not they were the homeowner and how many cars they had. Their response was homeowner: 14 no car or one car, two or more cars 86, not homeowner: 38 no car or one car, two or more cars 62. What percent of the families
  17. Theorem prove
    thales_1 We want to prove the sentence: If the natural number n is divisible by six, then n is divisible by three. From what assumption we started?