The Birthday Paradox - let's test it

[MagicMatt]

I learned about the Birthday Paradox in college (MATH 402 - Advanced Mathematical Statistics and Probability), and was recently discussing it with a coworker who didn't believe me. It was hard to confirm at work (our team is only about 15 people), so I want to put it to the test here.

For those who aren't familiar, the Birthday Paradox says that if you have a random group of 23 people, there is > 50% chance that 2 of these people share a birthday. Seems crazy at first, right? Check out the next post for details on the math.

To participate, it's real simple: Post your birth month & day (no year). Except if you were born on Leap Day. Sorry, you don't get to play.

We'll see how long it takes to get a repeating day. I'll take each consecutive group of 23 unique posters and note them here.

1st Duplicate: 16 people (03/16)
2nd Duplicate: 34 people (03/08)
3rd Duplicate: 39 people (01/05)
4th Duplicate: 45 people (02/06)
5th Duplicate: 53 people (08/15)
6th Duplicate: 56 people (10/30)
7th Duplicate: 58 people (08/13)
8th Duplicate: 60 people (06/03)
9th Duplicate: 62 people (12/11)
10th Duplicate: 65 people (08/16)
All 12 Months: 67 people (May finally accounted for)
11th Duplicate: 71 people (02/23)
12th Duplicate: 73 people (04/22)
13th Duplicate: 78 people (04/29)
1st Triplicate: 82 people (02/06)
14th Duplicate: 83 people (09/11)
15th Duplicate: 88 people (12/31)
16th Duplicate: 89 people (03/02)


TALLY

1. 01/01
2. 01/05
3. 01/05
4. 01/09
5. 01/13
6. 01/22
7. 01/23
8. 01/29
9. 01/31
10. 02/03
11. 02/06
12. 02/06
13. 02/06
14. 02/22
15. 02/23
16. 02/23
17. 02/26
18. 02/29
19. 03/02
20. 03/02
21. 03/04
22. 03/08
23. 03/08
24. 03/16
25. 03/16
26. 03/21
27. 04/01
28. 04/08
29. 04/09
30. 04/11
31. 04/22
32. 04/22
33. 04/24
34. 04/29
35. 04/29
36. 04/30
37. 05/05
38. 05/19
39. 06/01
40. 06/03
41. 06/03
42. 06/07
43. 06/12
44. 06/21
45. 06/26
46. 06/29
47. 07/07
48. 07/28
49. 08/03
50. 08/08
51. 08/13
52. 08/13
53. 08/15
54. 08/15
55. 08/16
56. 08/16
57. 08/18
58. 08/24
59. 08/28
60. 09/01
61. 09/11
62. 09/11
63. 09/15
64. 10/05
65. 10/11
66. 10/12
67. 10/16
68. 10/19
69. 10/24
70. 10/29
71. 10/30
72. 10/30
73. 10/31
74. 11/01
75. 11/05
76. 11/06
77. 11/08
78. 11/10
79. 11/17
80. 11/23
81. 11/24
82. 11/27
83. 11/30
84. 12/11
85. 12/11
86. 12/13
87. 12/19
88. 12/25
89. 12/31
90. 12/31

​

 

[MagicMatt]

The Birthday Paradox states that if you have 23 random people in a group, there's a >50% chance that two of them share a birthday. It also states that if there are 57 people in a group, the probability that two of them share a birthday is >99%.

Note that this does not include Feb. 29.

To see how it works, it's easier to find the converse probability (that no two people in the 23 person group share the same birthday). So we start at the bottom, first finding the probability that 2 people in the group don't have the same birthday.


Person 1 has a 365/365 chance of having a unique birthday (duh). So Person 2 has a 364/365 chance of having a unique birthday. Thus, the probability of two people in the group having unique birthdays is:

(365/365) x (364/365) = 99.72%

The probability that 3 people have unique birthdays is:

(365/365) x (364/365) x (363/365) = 99.17%

Four people....

(365/365) x (364/365) x (363/365) x (362/365) = 98.36%

So moving on, the probability that 23 people have unique birthdays is:

(365/365) x (364/365) x (363/365) x ... x (344/365) x (343/365) = 49.27%

That means that the probability that two of them do share a birthday is:

100% - 49.27% = [B]50.73%[/B]

Let's see how many people it takes us before we get a duplicate birthday. According the math it should happen with almost 100% certainty by the 58th person.

 

[MagicMatt]

I'll start: March 21

 

[jcav]

September 11

 

[jafo28]

June 21

 

[Chanoc]

August 28th

 

[vince805]

April 1

 

[jerbrew]

February 23

 

[mykey]

October 30

 

[JonM]

December 19

 

[AntonioMartins]

August 18

 

[Yooper]

March 8

 

[dsniegocki]

March 4

 

[shellsbells]

March 16

 

[KEG99]

January 22

 

[JDXX1971]

June 1

 

[smurfjuice]

Oct 11 ... happy berfday me ...

 

[CRZ]

March 16, Happy Birthday!!!

 

[aeviaanah]

June 29

 

[mattech]

Dec 31

 

[Wolfhound180]

Nov 10

 

[BigFloppy]

Apr 22

 

[NothingRhymesWithCurtiss]

Nov 1

 

[MagicMatt]

Nice, we hit the first duplicate in 16 people. Let's see when the next one happens, keep it going.....

 

[Zuljin]

10/19

It's not a paradox.
1. Leaving out the year opens the field a lot. Life expectancy is somewhere around 72 years.
2. About 350,000 people are born each day.
3. It's not about just my birthday. It's about each birthday in the group.

There are a fixed number of calendar days to be shared by an increasing number of people, which is well above that fixed number. There will be repeats.

 

[addled_mind]

Oct. 24

 

[TheFlatLander]

April 29

 

[gratus fermentatio]

Jan. 9th

 

[Moebbels]

Nov. 23

 

[Pkrd]

Nov 24

 

[Inkleg]

August 24

 

[ChuckO]

November 1

 

[chudsonvt]

April 8th

 

[Funkychef]

February 22nd

 

[jmcquesten]

Feb 3rd

 

[hedgie57]

April 24

 

[lschiavo]

January 5rd

 

[hockeybrewer]

3/8

 

[passedpawn]

August 15

 

[2eyejack]

April 11