[Guest]

At a movie theater, the manager announces that a free ticket will be given to the first person in line whose birthday is the same as someone in line who has already bought a ticket. You have the option of getting in line at any time. Assuming that you don't know anyone else's birthday, and that birthdays are uniformly distributed throughout a 365 day year, what position in line gives you the best chance of being the first duplicate birthday?

And what is the probability of getting a free ticket at the best position in line?

Edited: to add a followup question.

[Guest]

366 would guarantee a duplicate. However, that would also mean that you waited through 364 possible matches. According to the Birthday Paradox, the 50th person in line has approximately 97% chance of being a duplicate. After that, the line probability of a duplicate flattens significantly. It's *almost* guaranteed by the 100th person. Anyone after the 23rd person would have more than a 50% chance of being a match.

So, I haven't done the math, but you'd have to take the probability of the people in front of you not being a match AND you being a match that is the highest.

Just a guess, but I think it'd be somewhere in the high 30's.

[Guest]

At a movie theater, the manager announces that a free ticket will be given to the first person in line whose birthday is the same as someone in line who has already bought a ticket. You have the option of getting in line at any time. Assuming that you don't know anyone else's birthday, and that birthdays are uniformly distributed throughout a 365 day year, what position in line gives you the best chance of being the first duplicate birthday?

And what is the probability of getting a free ticket at the best position in line?

Edited: to add a follow up question.

I was told there would be no math!

Seriously, though, I think I disagree with the birtday paradox.

If you havae 23 people and compare birthday of 1 to the other 22 there are 22 chances.

If you compare each of them to each other there are 253 chances as 23.22/2 = 253.

However, am I mistaken by saying that by doing so compares 1 with 2 AND 2 with 1 which would create duplicates?

If so, then I would make the lowest number that covers more than half of the year to be 20.

With 2 people there is only 1 comparison

With 3 people there are 3 comparisons, 1-2, 1-3, 2-3

With 4 people there are 6 comparisons 1-2, 1-3, 1-4, 2-3, 2-4, 3-4

...

With 20 people there are 189 comparisons which covers more than half the year.

[Guest]

So far, people are missing the point to this question.

Let's ignore the February 29th issue for now.

1st in line: 0 chance of being a duplicate = 0.000%

2nd in line: 1/365 chance of being a duplicate = 0.274%

3rd in line: 1/365 chance of 2nd in line person winning, so 364/365 * 2/365 chance of winning = 0.546%

You have a MUCH better chance to win for being 3rd in line than 2nd. The question is, which position is the best?

20th spot in line. Probability of 0.0323.

I'll try to figure out the mathematical proof of it if I have time later.

[Guest]

So far, people are missing the point to this question.

Let's ignore the February 29th issue for now.

1st in line: 0 chance of being a duplicate = 0.000%

2nd in line: 1/365 chance of being a duplicate = 0.274%

3rd in line: 1/365 chance of 2nd in line person winning, so 364/365 * 2/365 chance of winning = 0.546%

You have a MUCH better chance to win for being 3rd in line than 2nd. The question is, which position is the best?

20th spot in line. Probability of 0.0323.

I'll try to figure out the mathematical proof of it if I have time later.

I SO love Excel.

And I'm glad someone else agrees with me

Any chance of sending that Excel sheet?

tanx

[Guest]

ADL          Pu                             Pnm                           Result                                        

364	0.997260274	0.002747253	0.27397%	                  2 People	99.72603%

363	0.994520548	0.002754821	0.82042%	                  3 People	99.17958%

362	0.991780822	0.002762431	1.63559%	                  4 People	98.36441%

361	0.989041096	0.002770083	2.71356%	                  5 People	97.28644%

360	0.98630137	0.002777778	4.04625%	                  6 People	95.95375%

359	0.983561644	0.002785515	5.62357%	                  7 People	94.37643%

358	0.980821918	0.002793296	7.43353%	                  8 People	92.56647%

357	0.978082192	0.00280112	9.46238%	                  9 People	90.53762%

356	0.975342466	0.002808989	11.69482%	10 People	88.30518%

355	0.97260274	0.002816901	14.11414%	11 People	85.88586%

354	0.969863014	0.002824859	16.70248%	12 People	83.29752%

353	0.967123288	0.002832861	19.44103%	13 People	80.55897%

352	0.964383562	0.002840909	22.31025%	14 People	77.68975%

351	0.961643836	0.002849003	25.29013%	15 People	74.70987%

350	0.95890411	0.002857143	28.36040%	16 People	71.63960%

349	0.956164384	0.00286533	31.50077%	17 People	68.49923%

348	0.953424658	0.002873563	34.69114%	18 People	65.30886%

347	0.950684932	0.002881844	37.91185%	19 People	62.08815%

346	0.947945205	0.002890173	41.14384%	20 People	58.85616%

345	0.945205479	0.002898551	44.36883%	21 People	55.63117%

344	0.942465753	0.002906977	47.56953%	22 People	52.43047%

343	0.939726027	0.002915452	50.72972%	23 People	49.27028%

342	0.936986301	0.002923977	53.83443%	24 People	46.16557%

341	0.934246575	0.002932551	56.86997%	25 People	43.13003%

340	0.931506849	0.002941176	59.82408%	26 People	40.17592%

339	0.928767123	0.002949853	62.68593%	27 People	37.31407%

338	0.926027397	0.00295858	65.44615%	28 People	34.55385%

337	0.923287671	0.002967359	68.09685%	29 People	31.90315%

336	0.920547945	0.00297619	70.63162%	30 People	29.36838%

335	0.917808219	0.002985075	73.04546%	31 People	26.95454%

334	0.915068493	0.002994012	75.33475%	32 People	24.66525%

333	0.912328767	0.003003003	77.49719%	33 People	22.50281%

332	0.909589041	0.003012048	79.53169%	34 People	20.46831%

331	0.906849315	0.003021148	81.43832%	35 People	18.56168%

330	0.904109589	0.003030303	83.21821%	36 People	16.78179%

329	0.901369863	0.003039514	84.87340%	37 People	15.12660%

328	0.898630137	0.00304878	86.40678%	38 People	13.59322%

327	0.895890411	0.003058104	87.82197%	39 People	12.17803%

326	0.893150685	0.003067485	89.12318%	40 People	10.87682%

325	0.890410959	0.003076923	90.31516%	41 People	9.68484%

324	0.887671233	0.00308642	91.40305%	42 People	8.59695%

323	0.884931507	0.003095975	92.39229%	43 People	7.60771%

322	0.882191781	0.00310559	93.28854%	44 People	6.71146%

321	0.879452055	0.003115265	94.09759%	45 People	5.90241%

320	0.876712329	0.003125	                94.82528%	46 People	5.17472%

319	0.873972603	0.003134796	95.47744%	47 People	4.52256%

318	0.871232877	0.003144654	96.05980%	48 People	3.94020%

317	0.868493151	0.003154574	96.57796%	49 People	3.42204%

316	0.865753425	0.003164557	97.03736%	50 People	2.96264%

315	0.863013699	0.003174603	97.44320%	51 People	2.55680%

314	0.860273973	0.003184713	97.80045%	52 People	2.19955%

313	0.857534247	0.003194888	98.11381%	53 People	1.88619%

312	0.854794521	0.003205128	98.38770%	54 People	1.61230%

311	0.852054795	0.003215434	98.62623%	55 People	1.37377%

310	0.849315068	0.003225806	98.83324%	56 People	1.16676%

309	0.846575342	0.003236246	99.01225%	57 People	0.98775%

308	0.843835616	0.003246753	99.16650%	58 People	0.83350%

307	0.84109589	0.003257329	99.29894%	59 People	0.70106%

306	0.838356164	0.003267974	99.41227%	60 People	0.58773%

305	0.835616438	0.003278689	99.50888%	61 People	0.49112%

304	0.832876712	0.003289474	99.59096%	62 People	0.40904%

303	0.830136986	0.00330033	99.66044%	63 People	0.33956%

302	0.82739726	0.003311258	99.71905%	64 People	0.28095%

301	0.824657534	0.003322259	99.76831%	65 People	0.23169%

300	0.821917808	0.003333333	99.80957%	66 People	0.19043%

299	0.819178082	0.003344482	99.84400%	67 People	0.15600%

298	0.816438356	0.003355705	99.87264%	68 People	0.12736%

297	0.81369863	0.003367003	99.89637%	69 People	0.10363%

296	0.810958904	0.003378378	99.91596%	70 People	0.08404%

295	0.808219178	0.003389831	99.93208%	71 People	0.06792%

294	0.805479452	0.003401361	99.94529%	72 People	0.05471%

293	0.802739726	0.003412969	99.95608%	73 People	0.04392%

292	0.8		0.003424658	99.96486%	74 People	0.03514%

291	0.797260274	0.003436426	99.97199%	75 People	0.02801%

290	0.794520548	0.003448276	99.97774%	76 People	0.02226%

289	0.791780822	0.003460208	99.98238%	77 People	0.01762%

288	0.789041096	0.003472222	99.98610%	78 People	0.01390%

287	0.78630137	0.003484321	99.98907%	79 People	0.01093%

286	0.783561644	0.003496503	99.99143%	80 People	0.00857%

285	0.780821918	0.003508772	99.99331%	81 People	0.00669%

284	0.778082192	0.003521127	99.99480%	82 People	0.00520%

283	0.775342466	0.003533569	99.99596%	83 People	0.00404%

282	0.77260274	0.003546099	99.99688%	84 People	0.00312%

281	0.769863014	0.003558719	99.99760%	85 People	0.00240%

280	0.767123288	0.003571429	99.99816%	86 People	0.00184%

279	0.764383562	0.003584229	99.99859%	87 People	0.00141%

278	0.761643836	0.003597122	99.99893%	88 People	0.00107%

277	0.75890411	0.003610108	99.99919%	89 People	0.00081%

276	0.756164384	0.003623188	99.99938%	90 People	0.00062%

275	0.753424658	0.003636364	99.99954%	91 People	0.00046%

274	0.750684932	0.003649635	99.99965%	92 People	0.00035%

273	0.747945205	0.003663004	99.99974%	93 People	0.00026%

272	0.745205479	0.003676471	99.99981%	94 People	0.00019%

271	0.742465753	0.003690037	99.99986%	95 People	0.00014%

270	0.739726027	0.003703704	99.99989%	96 People	0.00011%

269	0.736986301	0.003717472	99.99992%	97 People	0.00008%

268	0.734246575	0.003731343	99.99994%	98 People	0.00006%

267	0.731506849	0.003745318	99.99996%	99 People	0.00004%

266	0.728767123	0.003759398	99.99997%	100 People	0.00003%

265	0.726027397	0.003773585	99.99998%	101 People	0.00002%

264	0.723287671	0.003787879	99.99998%	102 People	0.00002%

263	0.720547945	0.003802281	99.99999%	103 People	0.00001%

262	0.717808219	0.003816794	99.99999%	104 People	0.00001%

261	0.715068493	0.003831418	99.99999%	105 People	0.00001%

260	0.712328767	0.003846154	100.00000%	106 People	0.00000%

[/codebox]

Explanation;

ADL = Available Unique Days Left

Pu = ADL/365

Pnm = 1/ADL, Showing the percent of of possibility getting a unique day

Result = The percent that you will draw a match

The formula is =1-Product($B$2:Cell in B)

Edited April 18, 2008 by PolishNorbi

[Guest]

More explanation on my table since it posted prematurily..

Basically here is what I did;

Column A, Amount of days left without a matching b-day.

Column B, Column A/365 - This shows the probability of drawing the days. IE; if you are the 366 person, you have a literal 0/365 chance of drawing a unique day.

Column C, Forgot what this was for exactly, its not part of any formula its just 1/Available days left.

Column D, Its the probability for all the people including you drawing no match subtracted from 1.

IE;

Person 2, has a 364/365 chance of drawing no match.

Person 3, has a 363/365 chance of drawing no match if Person had his 364/365 chance.

So the probability for Person 3 of not drawing a match is 364/365 * 363/365

To get the result of a match subtract that from 1.

That is what in D.

Column E, to show the result for what person in line.

Column F, shows the probability of drawing a non-match

Edited April 18, 2008 by PolishNorbi

[Guest]

I SO love Excel.

And I'm glad someone else agrees with me

Any chance of sending that Excel sheet?

tanx

[spoiler='I Don't

']Reasoning:

If there are 3 people in line, there is only 2 comparisons.

3 with 2 and 1.

Comparisons 2 with 1 have already been done when 2 arrived.

When 4 arrives, we compare him with 3, 2, and 1 if 3 didn't match 2 or 1 and if 2 didn't match 1.

So that means it would be The probability of 2 being unique from 1 (364/365) * The probability of 3 being unique from 2,1 (363/365) * the probability of you not being a match (362/365)

This is the probability of you not matching, but for you to match it would the 1- "the above percent". That is what I used to calculate

Another simpler way to think of the above (since I got confused myself explaining) is this;

What if we look at 5: Well for that to happen, 4 couldnt match 3,2,1 and 3 couldn't match 2,1 and 2 couldn't match 1.

If you sum up those probabiliies (4 not match....) then you get the probability that 5 has at having a chance. 100% - that percent = the percent that 5 is already too late. That same percent is the percent 4 matched.

Edit: Simple explanation added and grammer

Edited April 18, 2008 by PolishNorbi

[Guest]

At a movie theater, the manager announces that a free ticket will be given to the first person in line whose birthday is the same as someone in line who has already bought a ticket. You have the option of getting in line at any time. Assuming that you don't know anyone else's birthday, and that birthdays are uniformly distributed throughout a 365 day year, what position in line gives you the best chance of being the first duplicate birthday?

And what is the probability of getting a free ticket at the best position in line?

Edited: to add a follow up question.

I'll go postition 11

so long as there are no Lyrian III aliens in the equaion

Edited to add silly comment!

Edited April 18, 2008 by Lost in space

[Guest]

benkenobi and rhapsodize have got it. Well done guys.

I have posted my excel solution. If someone interested about the mathematical solution, please feel free to PM me or post your comments. Here we go.

Line No.	Propability of 	         Probability of none of 	Probability of getting 

                    sharing BD             n-1 share BD                     free ticket

  	            (A3-1)/365	          (365-(A3-2))/365*C2	            B3*C3               <---------(third row)

       2	   0.00274	                  1.00000	               0.00274

       3	   0.00548	                  0.99726	               0.00546

       4	   0.00822	                  0.99180	               0.00815

       5	   0.01096                        0.98364	               0.01078

       6	   0.01370	                  0.97286	               0.01333

       7	   0.01644	                  0.95954	               0.01577

       8	   0.01918	                  0.94376	               0.01810

       9	   0.02192	                  0.92566	               0.02029

      10	  0.02466	                 0.90538	              0.02232

      11	  0.02740	                 0.88305	              0.02419

      12	  0.03014	                 0.85886	              0.02588

      13	  0.03288	                 0.83298	              0.02739

      14	  0.03562	                 0.80559	              0.02869

      15	  0.03836	                 0.77690	              0.02980

      16	  0.04110	                 0.74710	              0.03070

      17	  0.04384	                 0.71640	              0.03140

      18	  0.04658	                 0.68499	              0.03190

      19	  0.04932	                 0.65309	              0.03221

[b]      20	  0.05205	                 0.62088	              0.03232[/b]

      21	  0.05479	                 0.58856	              0.03225

      22	  0.05753	                 0.55631	              0.03201

      23	  0.06027	                 0.52430	              0.03160

      24	  0.06301	                 0.49270	              0.03105

      25	  0.06575	                 0.46166	              0.03036

      26	  0.06849	                 0.43130	              0.02954

      27	  0.07123	                 0.40176	              0.02862

      28	  0.07397	                 0.37314	              0.02760

      29	  0.07671	                 0.34554	              0.02651

      30	  0.07945	                 0.31903	              0.02535

      31	  0.08219	                 0.29368	              0.02414

      32	  0.08493	                 0.26955	              0.02289

      33	  0.08767	                 0.24665	              0.02162

      34	  0.09041	                 0.22503	              0.02035

      35	  0.09315	                 0.20468	              0.01907

      ...

[/codebox]

[Guest]

I would be 2nd in line and stand behind my twin sister... (if I had one...) which would give me 100% probability... YAY! Sorry, I'm not good at these... But I like the answers I have seen so far.

[Guest]

Changing my answer

from 11 to 55

It depends on the month and day. Birth rates are actually seasonal. More people are born in August than any other month. In a group of 23 people, the odds are 50-50 that 2 of them will share a birthday. For a group of 30 people, the odds go up to 71% that 2 people will have the same birthday.

actually i could stand behind my daughter as we share the same birthday! yeah free ticket for me position 1 n 2

Edited April 19, 2008 by Lost in space