Lets test this out using a no science or mathematics what so ever. Using
http://www.random.org as the selection for each of the following
Lets consider for the "Same Place Everytime" scenario s/he enters 1st
1, 5, 8, 3, 7, 3, 9, 7, 1, 9
1st appears twice, 9th appears twice and 7th appears twice. This would of course gain him/her 2 wins out of 10. Not bad by any stretch of the word but certainly not most of the time.
Now for our second test lets use the results from the last test and pump out 10 more numbers from random.org.
7, 8, 3, 3, 6, 7, 7, 1, 2, 4
In this situation s/he would have only won once with a pair of 3's matching. In an ideal world this is how it'd work out. However as you can see the number 7 appears 3 times and the number 3 appears twice. Again certainly not most of the time but that's assume he'd follow the working of a RNG himself. With bigger ranges this pattern should be hopefully be non existent but may still appear from time to time. Unlike lets say the lottery the numbers don't get chosen and then not used again for the duration of that drawing.
For reading pleasure here is some sample data for a few higher data ranges
11, 5, 10, 2, 15, 1, 12, 11, 6, 3
9, 4, 3, 9, 5, 1, 4, 7, 10, 9, 7, 9, 1, 1, 1
And, yes, 1 did really appear 3 times one after the other.
Anyway, I've probably complicated matters further so with this long post, I take my leave