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=13650.0027  q2=1364365=13650.0027 q3=1(1q2) 363365=1(10.0027) 3633650.0082 q4=1(1q3) 362365=1(10.0082) 3623650.0164  q5=1(1q4) 3654365=1(10.0164) 36543650.0271 q6=1(1q5) 3655365=1(10.0271) 36553650.0405 q7=1(1q6) 3656365=1(10.0405) 36563650.0562 q8=1(1q7) 3657365=1(10.0562) 36573650.0743 q9=1(1q8) 3658365=1(10.0743) 36583650.0946 q10=1(1q9) 3659365=1(10.0946) 36593650.1169 q11=1(1q10) 36510365=1(10.1169) 365103650.1411 q12=1(1q11) 36511365=1(10.1411) 365113650.167 q13=1(1q12) 36512365=1(10.167) 365123650.1944 q14=1(1q13) 36513365=1(10.1944) 365133650.2231 q15=1(1q14) 36514365=1(10.2231) 365143650.2529 q16=1(1q15) 36515365=1(10.2529) 365153650.2836 q17=1(1q16) 36516365=1(10.2836) 365163650.315 q18=1(1q17) 36517365=1(10.315) 365173650.3469 q19=1(1q18) 36518365=1(10.3469) 365183650.3791 q20=1(1q19) 36519365=1(10.3791) 365193650.4114 q21=1(1q20) 36520365=1(10.4114) 365203650.4437 q22=1(1q21) 36521365=1(10.4437) 365213650.4757 q23=1(1q22) 36522365=1(10.4757) 365223650.5073 q24=1(1q23) 36523365=1(10.5073) 365233650.5383 q25=1(1q24) 36524365=1(10.5383) 365243650.5687 q26=1(1q25) 36525365=1(10.5687) 365253650.5982 q27=1(1q26) 36526365=1(10.5982) 365263650.6269 q28=1(1q27) 36527365=1(10.6269) 365273650.6545 q29=1(1q28) 36528365=1(10.6545) 365283650.681 q30=1(1q29) 36529365=1(10.681) 365293650.7063 q31=1(1q30) 36530365=1(10.7063) 365303650.7305 q32=1(1q31) 36531365=1(10.7305) 365313650.7533 q33=1(1q32) 36532365=1(10.7533) 365323650.775 q34=1(1q33) 36533365=1(10.775) 365333650.7953 q35=1(1q34) 36534365=1(10.7953) 365343650.8144 q36=1(1q35) 36535365=1(10.8144) 365353650.8322 q37=1(1q36) 36536365=1(10.8322) 365363650.8487 q38=1(1q37) 36537365=1(10.8487) 365373650.8641 q39=1(1q38) 36538365=1(10.8641) 365383650.8782 q40=1(1q39) 36539365=1(10.8782) 365393650.8912 q41=1(1q40) 36540365=1(10.8912) 365403650.9032 q42=1(1q41) 36541365=1(10.9032) 365413650.914 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



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?

Next similar math problems:

  1. Car crash
    car1_7 On the road, with a maximum permitted speed of 60 km/h, there was a car crash. From the length of the vehicle's braking distance, which was 40 m, the police investigated whether the driver did not exceed that speed. What is the conclusion of the police, a
  2. A car
    car_31 A car weighing 1.05 tonnes driving at the maximum allowed speed in the village (50 km/h) hit a solid concrete bulkhead. Calculate height it would have to fall on the concrete surface to make the impact intensity the same as in the first case!
  3. Angled cyclist turn
    cyclistTurn The cyclist passes through a curve with a radius of 20 m at 25 km/h. How much angle does it have to bend from the vertical inward to the turn?
  4. Orlík hydroelectric plant
    1280px-Orlík_1 The Orlík hydroelectric power plant, built in 1954-1961, consists of four Kaplan turbines. For each of them, the water with a flow rate of Q = 150 m3/s is supplied with a flow rate of h = 70.5 m at full power. a) What is the total installed power of the
  5. Pump
    pumpa What power has a pump output to move 4853 hl of water to a height of 31 m for 8 hours?
  6. Cu thief
    trolleywire_1 The thief stole 121 meters copper wire with cross-section area of 103 mm2. Calculate how much money gets in the scrap redemption, if redeemed copper for 4.6 eur/kg? The density of copper is 8.96 t/m3.
  7. Copper sheet
    cuplech The copper plate has a length of 1 m, width 94 cm and weighs 9 kg. What is the plate thickness, if 1 m3 weighs 8715 kg?
  8. The water tank
    hydroglobus The water tank has the shape of a sphere with a radius of 2 m. How many liters of water will fit in the tank? How many kilograms of paint do we need to paint the tank, if we paint with 1 kg of paint 10 m2?
  9. The copper wire
    cu_wire The copper wire bundle with a diameter of 2.8mm has a weight of 5kg. How many meters of wire is bundled if 1m3 of copper weighs 8930kg?
  10. Heptagonal pyramid
    truncated_hexagonal_pyramid A hardwood for a column is in the form of a frustum of a regular heptagonal pyramid. The lower base edge is 18 cm and the upper base of 14 cm. The altitude is 30 cm. Determine the weight in kg if the density of the wood is 10 grams/cm3.
  11. Friction coefficient
    car What is the weight of a car when it moves on a horizontal road at a speed of v = 50 km/h at engine power P = 7 kW? The friction coefficient is 0.07
  12. Iron density
    pipe1_2 Calculate the weight of a 2 m long rail pipe with an internal diameter of 10 cm and a wall thickness of 3 mm. The iron density is p = 7.8 g/cm3.
  13. Water level
    bazen_11 How high is the water in the swimming pool with dimensions of 37m in length and 15m in width, if an inlet valve is opened for 10 hours flowing 12 liters of water per second?
  14. Density of the concrete
    beton_1 Find the density of the concrete of the cuboid-shaped column has dimensions of 20 x 20 cm x 2 m if the weight of the column is 200 kg.
  15. 3d printer
    filament 3D printing ABS filament with diameter 1.75 mm has density 1.04 g/cm3. Find the length of m = 5 kg spool filament. (how to calculate length)
  16. Iron pole
    ministranti What is the mass of pole with the shape of a regular quadrilateral prism with a length of 1 m and a cross-sectional side length of a = 4.5 cm make from iron with density ρ = 7800 kg/m³?
  17. Children pool
    hexagon_prism2 The bottom of the children's pool is a regular hexagon with a = 60 cm side. The distance of opposing sides is 104 cm, the height of the pool is 45 cm. A) How many liters of water can fit into the pool? B) The pool is made of a double layer of plastic film