Question 287820
The probability that none of the numbers are the same is equal to:


{{{49/49 * 48/49 * 47/49 * 46/49 * 45/49 * 44/49}}} = {{{.727414103}}}.


The probability that at least two of them will be the same is equal to 1 minus the probability that none are the same.


This equals {{{1 - .727414103}}} = {{{.272585897}}}.


To see how this works, try the same type of problem with smaller numbers.


Assume the numbers in the pot are [1,2,3,4].


Assume you want to draw 3 times.


What is the probability that none of the numbers are the same?


That probability would be {{{4/4 * 3/4 * 2/4}}} = {{{24/64}}} = {{{3/8}}}.


On your first draw, you can get 4 out of 4 = 4/4.


On your second draw, you can get 3 out of 4 = 3/4.   That's because once you picked a number, you do not want to pick it again, so the remaining choices are 3 out of 4.


On your third draw, you can get 2 out of 4 = 2/4.   That's because once you picked 2 numbers, you do not want to pick either of those again, so the remaining choices are 2 out of 4.


The probability that you will not get any number being the same as any other number is therefore:


{{{4/4 * 3/4 * 2/4}}} = {{{24/64}}} = {{{3/8}}}.


The probability that you will pick at least 2 numbers that are the same will be equal to 1 minus the probability of none being the same which will equal {{{1 - 3/8}}} = {{{5/8}}}.


To see how this works, we'll detail out the possibilities.


The numbers in the pot are [1,2,3,4].


you will draw 3 from the pot randomly.


The pot is always the same on each draw.


The possible sets of 3 you can get are:


111
112
113
114
121
122
123
124
131
132
133
134
141
142
143
144
211
212
213
214
221
222
223
224
231
232
233
234
241
242
243
244
311
312
313
314
321
322
323
324
331
332
333
334
341
342
343
344
411
412
413
414
421
422
423
424
431
432
433
434
441
442
443
444


That's a total of 64 possible sets of 3 you can draw from a pot of 4 consecutive numbers.


Out of that 64, the sets where all the numbers are different are:


123
124
132
134
142
143
213
214
231
234
241
243
312
314
321
324
341
342
412
413
421
423
431
432


That's a total of 24 out of 64.


Out of that 64, the sets where at least 2 of the numbers are the same are:


111
112
113
114
121
122
131
133
141
144
211
212
221
222
223
224
232
233
242
244
311
313
322
323
331
332
333
334
343
344
411
414
422
424
433
434
441
442
443
444


That's a total of 40 out of 64


The method that is good for drawing a set of 3 out of a set of 4 will be the same as the method for drawing a set of 6 out of a set of 49.


The method is:


To find the probability that at least 2 of the numbers are the same, you find the probability that none of the numbers are the same and you subtract that from 1.


When you draw a set of 3 out of 4 possible numbers, the probability that at least 2 of the numbers are the same will be:


{{{1 - ((4*3*2)/(4*4*4))}}}


That becomes:


{{{1 - (24/64)}}} which reduces to {{{1 - (3/8)}}} which equals {{{5/8}}}.


When you draw a set of 6 out of 49 possible numbers, the probability that at least 2 of the number are the same will be:


{{{1 - ((49*48*47*46*45*44)/(49*49*49*49*49*49))}}}


That becomes:


{{{1 - ((1.006834752*10^10)/(1.38412872*10^10))}}} which becomes {{{1 - (1.006834752/1.38412872)}}} which becomes {{{1 - .727414103}}} which equals {{{.272585897}}}.