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. The coal
    coal The coal stock would be enough to heat a larger room for 12 weeks, a smaller one for 18 weeks. It was heated for four weeks in both rooms, then only in a smaller one. How long was the coal stock enough?
  2. Cyclists and walkers
    time_1 A group of tourists started at 8:00 at speed 4 km/h walk. At half-past ten, another group started on a bike and caught up with a group of tourists at 10:50. What was the average speed of cyclists?
  3. Water current
    river John swims upstream. After a while, he passes the bottle, from that moment he floats for 20 minutes in the same direction. He then turns around and swims back, and from the first meeting with the bottle, he sails 2 kilometers before he reaches the bottle.
  4. Resistance of the resistor
    Elektrárna_1 The resistor terminals have a voltage of 20 V and a current of 5 mA is passed through. What is the resistance of the resistor?
  5. Transformer
    trafo Solve the textbook problems - transformer: a) N1 = 40, N2 = 80, U2 = 80 V, U1 =? b) N1 = 400, U1 = 200 V, U2 = 50 V, N2 =?
  6. Filament of bulb
    bulb _1 The filament of bulb has a 1 ohm resistivity and is connected to a voltage 220 V. How much electric charge will pass through the fiber when the electric current passes for 10 seconds?
  7. Coil as a girl
    tlmivka_TL1.JPG The electrical resistance of the copper wire coil is 2.0 ohms. What current runs through the coil when the voltage between the terminals is 3.0 V?
  8. Resistance
    bulb_1 Determine the resistance of the bulb with current 200 mA and is in regular lamp (230V).
  9. Fog
    fog The car started in fog at speed 30 km/h. After a 12-minute drive, the fog dissipated and the driver drove next 12 minutes distance 17 km. On the last 17 km long again the driving conditions deteriorated and the driver drove the speed of 51 km/h. a) Calc
  10. Copper Cu wire
    cu_wire Copper wire with a diameter of 1 mm and a weight of 350 g is wound on a spool. Calculate its length if the copper density is p = 8.9 g/cm cubic.
  11. Closed circuit
    voltemeter In a closed circuit, there is a voltage source with U1 = 12 V and with an internal resistance R1 = 0.2 Ω. The external resistance is R2 = 19.8 Ω. Determine the electric current and terminal voltage.
  12. Cooker
    kWMeter A current of 2A passes through the immersion cooker at a voltage of 230V. What work do the electric field forces in 2 minutes?
  13. 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?
  14. The shooter
    terc2 The shooter heard the impact of the bullet on the target in one second after the shot. The bullet was moving at an average speed of 500 m/s. Calculate with speed of sound of 340 m/s. Determine the distance of the target.
  15. Aircraft angines
    aircraft-02 The two engines of the aircraft are enough to supply the fuel for five hours of operation. However, one of the engines has a malfunction and thus consumes one-third more fuel. How long can the plane be in the air before it runs out of fuel? After an hour
  16. Cheetah vs antelope
    motion2 When the cheetah began chasing the antelope, the distance between them was 120 meters. Although the antelope was running at 72km/h, the cheetah caught up with it in 12 seconds. What speed was the cheetah running?
  17. Where and when
    cars The truck left Kremnica at 11:00 h at a speed of 60km/h. At 12:30 h, the passenger car started at an average speed of 80km/h. How many kilometers from Kremnica will the passenger car reach truck, and when?