Let k equal the number of people.
The probability that none of these k people have the same birthday is:
This solution does not take into account people that were born on February 29. And, this solution assumes that in the population of interest, an equal number of people have birthdays on each of the 365 days of the year, and the group of people at hand represents a random sample from that population.
Here are the results. If you take (1 - the probability displayed), that's the probability that there is AT LEAST one duplicate birthday in the group.
Number of People and the Probability None Have the Same Birthday
2people 0.99726027397260
3people 0.99179583411522
4people 0.98364408753345
5people 0.97286442630021
6people 0.95953751635089
7people 0.94376429690402
8people 0.92566470764833
9people 0.90537616611083
10people 0.88305182228892
11people 0.85885862167827
12people 0.83297521116194
13people 0.80558972476757
14people 0.77689748799503
15people 0.74709868023631
16people 0.71639599474715
17people 0.68499233470344
18people 0.65308858212821
19people 0.62088147396846
20people 0.58856161641942
21people 0.55631166483479
22people 0.52430469233745
23people 0.49270276567602
24people 0.46165574208547
25people 0.43130029603054
25people 0.40175917986406
27people 0.37314071773676
28people 0.34553852765760
29people 0.31903146252222
30people 0.29368375728073
31people 0.26954536627136
32people 0.24665247214968
33people 0.22502814582423
34people 0.20468313537985
35people 0.18561676112529
36people 0.16781789362012
37people 0.15126599178361
38people 0.13593217891788
39people 0.12178033563328
40people 0.10876819018205
41people 0.09684838851827
42people 0.08596952843813
43people 0.07607714434388
44people 0.06711463144857
45people 0.05902410053423
46people 0.05174715663274
47people 0.04522559716670
48people 0.03940202712058
49people 0.03422039067732
50people 0.02962642042201
51people 0.02556800666557
52people 0.02199549066572
53people 0.01886188651609
54people 0.01612303724115
55people 0.01373771118355
56people 0.01166764511480
57people 0.00987754065883
58people 0.00833502061074
59people 0.00701055158218
60people 0.00587733913465
61people 0.00491120119471
62people 0.00409042510463
63people 0.00339561316905
64people 0.00280952103302
65people 0.00231689268751
66people 0.00190429535960
67people 0.00155995702060
68people 0.00127360874559
69people 0.00103633369161
70people 0.00084042403484
71people 0.00067924682268
72people 0.00054711935854
73people 0.00043919444398
74people 0.00035135555519
75people 0.00028012182619
76people 0.00022256254683
77people 0.00017622075626
78people 0.00013904541864
79people 0.00010933160315
80people 0.00008566805069
81people 0.00006689149163
82people 0.00005204707842
83people 0.00004035431012
84people 0.00003117785056
85people 0.00002400267399
86people 0.00001841301018
87people 0.00001407460231
88people 0.00001071983408
89people 0.00000813532614
90people 0.00000615164388
91people 0.00000463480018
92people 0.00000347927466
93people 0.00000260230680
94people 0.00000193925328
95people 0.00000143982915
96people 0.00000106507910
97people 7.84948705257e-7
98people 5.76345898673e-7
99people 4.21600972467e-7
100people 3.07248927855e-7
101people 2.23071139449e-7
102people 1.61344605010e-7
103people 1.16256523630e-7
104people 8.34498882218e-8
105people 5.96723858104e-8
106people 4.25063569898e-8
107people 3.01620450838e-8
108people 2.13200208599e-8
109people 1.50116311604e-8
110people 1.05287056851e-8
111people 7.35567107224e-9
112people 5.11874098663e-9
113people 3.54805884673e-9
114people 2.44961873008e-9
115people 1.68453229143e-9
116people 1.15378928633e-9
117people 7.8710560292e-10
118people 5.3480053719e-10
119people 3.6190617170e-10
120people 2.4391488829e-10
121people 1.6372370126e-10
122people 1.0944822826e-10
123people 7.2865491418e-11
124people 4.8310799805e-11
125people 3.1898372832e-11
126people 2.0974222359e-11
127people 1.3733791882e-11
128people 8.9551699389e-12
129people 5.8146820692e-12
130people 3.7596592506e-12
131people 2.4206192606e-12
132people 1.5518697438e-12
133people 9.9065200487e-13
134people 6.2971849957e-13
135people 3.9857006584e-13
136people 2.5113244817e-13
137people 1.5754064719e-13
138people 9.8365759982e-14
139people 6.1173288657e-14
140people 3.7858605140e-14
141people 2.3314683517e-14
142people 1.4321877018e-14
143people 8.7707618945e-15
144people 5.3290705182e-15
145people 3.2266426973e-15
146people 1.9448257354e-15
147people 1.1668954412e-15
148people 6.9694029092e-16
149people 4.1434532364e-16
150people 2.4520161618e-16
151people 1.4443382871e-16
152people 8.4681751628e-17
153people 4.9417022183e-17
154people 2.8702489597e-17
155people 1.6592398096e-17
156people 9.5463112331e-18
157people 5.4662439663e-18
158people 3.1150102603e-18
159people 1.7665948599e-18
160people 9.9703709903e-19
161people 5.5997974055e-19
162people 3.1297497828e-19
163people 1.7406553587e-19
164people 9.6332159575e-20
165people 5.3048668697e-20
166people 2.9067763670e-20
167people 1.5847904028e-20
168people 8.5969451989e-21
169people 4.6399950799e-21
170people 2.4916137963e-21
171people 1.3311361378e-21
172people 7.0750797460e-22
173people 3.7410695643e-22
174people 1.9679050859e-22
175people 1.0297804696e-22
176people 5.3605010746e-23
177people 2.7757115154e-23
178people 1.4296815476e-23
179people 7.3246698468e-24
180people 3.7325714836e-24
181people 1.8918512999e-24
182people 9.5370038132e-25
183people 4.7815662954e-25
184people 2.3842330569e-25
185people 1.1823183104e-25
186people 5.8306108458e-26
187people 2.8593954559e-26
188people 1.3944449072e-26
189people 6.7621027009e-27
190people 3.2606303434e-27
191people 1.5633159181e-27
192people 7.4525197191e-28
193people 3.5322901682e-28
194people 1.6645312574e-28
195people 7.7982149317e-29
196people 3.6320453107e-29
197people 1.6816867329e-29
198people 7.7403663322e-30
199people 3.5414826780e-30
200people 1.6106469166e-30
201people 7.2810066092e-31
202people 3.2714659833e-31
203people 1.4609560419e-31
204people 6.4842432543e-32
205people 2.8601730519e-32
206people 1.2537744885e-32
207people 5.4616477718e-33
208people 2.3642201314e-33
209people 1.0169385223e-33
210people 4.3463673828e-34
211people 1.8457176557e-34
212people 7.7874114789e-35
213people 3.2643122090e-35
214people 1.3593848103e-35
215people 5.6237563385e-36
216people 2.3111327419e-36
217people 9.4344870832e-37
218people 3.8254906529e-37
219people 1.5406770575e-37
220people 6.1627082299e-38
221people 2.4481991598e-38
222people 9.6586487401e-39
223people 3.7840733420e-39
224people 1.4721600399e-39
225people 5.6869744008e-40
226people 2.1813052496e-40
227people 8.3068884848e-41
228people 3.1406866052e-41
229people 1.1788330546e-41
230people 4.3923642581e-42
231people 1.6245730818e-42
232people 5.9641861084e-43
233people 2.1732513765e-43
234people 7.8594296355e-44
235people 2.8207815952e-44
236people 1.0046619380e-44
237people 3.5507230138e-45
238people 1.2451850569e-45
239people 4.3325617048e-46
240people 1.4956240406e-46
241people 5.1220001389e-47
242people 1.7400767595e-47
243people 5.8638203129e-48
244people 1.9599618580e-48
245people 6.4974078033e-49
246people 2.1361340723e-49
247people 6.9643823180e-50
248people 2.2514989411e-50
249people 7.2171335922e-51
250people 2.2936643745e-51
251people 7.2266137827e-52
252people 2.2570793732e-52
253people 6.9876703883e-53
254people 2.1441618726e-53
255people 6.5206018591e-54
256people 1.9651128890e-54
257people 5.8684193125e-55
258people 1.7364090020e-55
259people 5.0902948827e-56
260people 1.4782774180e-56
261people 4.2525788737e-57
262people 1.2116937065e-57
263people 3.4193000484e-58
264people 9.5553042448e-59
265people 2.6440704897e-59
266people 7.2440287388e-60
267people 1.9648187538e-60
268people 5.2754037774e-61
269people 1.4019566203e-61
270people 3.6873379602e-62
271people 9.5971809923e-63
272people 2.4716027761e-63
273people 6.2975084432e-64
274people 1.5873171966e-64
275people 3.9574209560e-65
276people 9.7580242751e-66
277people 2.3793538643e-66
278people 5.7365243853e-67
279people 1.3673359494e-67
280people 3.2216682643e-68
281people 7.5025151359e-69
282people 1.7266062231e-69
283people 3.9262552470e-70
284people 8.8206282261e-71
285people 1.9574544830e-71
286people 4.2903111957e-72
287people 9.2858790264e-73
288people 1.9843796276e-73
289people 4.1862255157e-74
290people 8.7165243614e-75
291people 1.7910666496e-75
292people 3.6312036184e-76
293people 7.2624072367e-77
294people 1.4325844412e-77
295people 2.7866711048e-78
296people 5.3443007490e-79
297people 1.0102924704e-79
298people 1.8821887119e-80
299people 3.4549765396e-81
300people 6.2473548388e-82
301people 1.1125426425e-82
302people 1.9507597020e-83
303people 3.3670646911e-84
304people 5.7193975574e-85
305people 9.5584452330e-86
306people 1.5712512712e-86
307people 2.5398308219e-87
308people 4.0358955526e-88
309people 6.3026314109e-89
310people 9.6697906578e-90
311people 1.4570917430e-90
312people 2.1556973731e-91
313people 3.1301907062e-92
314people 4.4594497732e-93
315people 6.2310120119e-94
316people 8.5356328930e-95
317people 1.1458794843e-95
318people 1.5069100067e-96
319people 1.9404046662e-97
320people 2.4454414971e-98
321people 3.0149278731e-99
322people 3.634433600e-100
323people 4.281661502e-101
324people 4.926843372e-102
325people 5.534262418e-103
326people 6.064945116e-104
327people 6.480352315e-105
328people 6.746668164e-106
329people 6.839088276e-107
330people 6.745402135e-108
331people 6.468193828e-109
332people 6.025166854e-110
333people 5.447411128e-111
334people 4.775812496e-112
335people 4.056169517e-113
336people 3.333837959e-114
337people 2.648802762e-115
338people 2.031958283e-116
339people 1.503092429e-117
340people 1.070695977e-118
341people 7.333534086e-120
342people 4.822049810e-121
343people 3.038551935e-122
344people 1.831455961e-123
345people 1.053714388e-124
346people 5.773777471e-126
347people 3.005527999e-127
348people 1.482178191e-128
349people 6.903295684e-130
350people 3.026102218e-131
351people 1.243603651e-132
352people 4.769986607e-134
353people 1.698899340e-135
354people 5.585422486e-137
355people 1.683278010e-138
356people 4.611720574e-140
357people 1.137136580e-141
358people 2.492354148e-143
359people 4.779857270e-145
360people 7.857299621e-147
361people 1.076342414e-148
362people 1.179553330e-150
363people 9.694958879e-153
364people 5.312306235e-155
365people 1.455426366e-157
366people 0