Birthday Paradox

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Let's consider there are 365 days in a year (sometime it has 366 but we will not consider that case). We want to find out how big a group needs to be such that there is a 50% chance that two people in this group have the same birthday.

You may think that well it needs about half of 365 people (which is about 183) people in the group. Well the actually is acutally much smaller than that.

To analyze this problem, let's start with a smaller group of people, for example 3 people. The probability that in the group of 3 people, at least two people has the same birthday is:

\[ 1-\frac{{}_{365} {P}_{3}}{365^3} \approx 0.008204 \]

\(\frac{{}_{365} {P}_{3}}{365^3}\) is the probability that all 3 people in the group has different birthday. 1 minus that gives the probability that at least two people in that group share the same birthday.

Using this method, we can find the probability that at least two people has the same birthday in a n people group (where \(2\leq n \leq 365\)) is:

\[ 1-\frac{{}_{365} {P}_{n}}{365^n} \]

The table below show the number of people in a group and the probability that at least two people at the corresponding group has the same birthday:

Number of People in a Group	The Probability that at Least Two People at the Group has the Same Birthday	Number of People in a Group	The Probability that at Least Two People at the Group has the Same Birthday
2	0.00273973	59	0.99298945
3	0.00820417	60	0.99412266
4	0.01635591	61	0.9950888
5	0.02713557	62	0.99590957
6	0.04046248	63	0.99660439
7	0.0562357	64	0.99719048
8	0.07433529	65	0.99768311
9	0.09462383	66	0.9980957
10	0.11694818	67	0.99844004
11	0.14114138	68	0.99872639
12	0.16702479	69	0.99896367
13	0.19441028	70	0.99915958
14	0.22310251	71	0.99932075
15	0.25290132	72	0.99945288
16	0.28360401	73	0.99956081
17	0.31500767	74	0.99964864
18	0.34691142	75	0.99971988
19	0.37911853	76	0.99977744
20	0.41143838	77	0.99982378
21	0.44368834	78	0.99986095
22	0.47569531	79	0.99989067
23	0.50729723	80	0.99991433
24	0.53834426	81	0.99993311
25	0.5686997	82	0.99994795
26	0.59824082	83	0.99995965
27	0.62685928	84	0.99996882
28	0.65446147	85	0.999976
29	0.68096854	86	0.99998159
30	0.70631624	87	0.99998593
31	0.73045463	88	0.99998928
32	0.75334753	89	0.99999186
33	0.77497185	90	0.99999385
34	0.79531686	91	0.99999537
35	0.81438324	92	0.99999652
36	0.83218211	93	0.9999974
37	0.84873401	94	0.99999806
38	0.86406782	95	0.99999856
39	0.87821966	96	0.99999893
40	0.89123181	97	0.99999922
41	0.90315161	98	0.99999942
42	0.91403047	99	0.99999958
43	0.92392286	100	0.99999969
44	0.93288537	101	0.99999978
45	0.9409759	102	0.99999984
46	0.94825284	103	0.99999988
47	0.9547744	104	0.99999992
48	0.96059797	105	0.99999994
49	0.96577961	106	0.99999996
50	0.97037358	107	0.99999997
51	0.97443199	108	0.99999998
52	0.97800451	109	0.99999998
53	0.98113811	110	0.99999999
54	0.98387696	111	0.99999999
55	0.98626229	112	0.99999999
56	0.98833235	113~364	\( \approx 1 \)
57	0.99012246	365	1
58	0.99166498	> 365	1

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