Duplicates in Multiple Selections

This section aims to prove that duplicates or choosing the same numbers more than once results in lower chances of winning.

First lets consider the case where there are no duplicates. For simplicity lets just consider taking two selections on a lotto ticket. I make sure that all my numbers are different. My two selections do not have any numbers in common.

Lets say I choose 

first selection : 1 2 3 4 5 6
2nd selection   : 7 8 9 10 11 12

Now I have two chances of getting all six correct. One chance for each selection. Thus there are two ways I can win. I call these winning combinations.

To get exactly five correct then five of the numbers from the first selection must come in or five winning numbers from the second selection.

There are 204 winning combinations for the first selection. There are 204 winning combinations for the second selection. The selections DO NOT have any winning combinations in common!! Therefore there are 2 * 204 = 408 winning combinations. That is 408 chances of winning without duplicates.

The problem occurs when we have selections with common elements. A common element is one that occurs in both selections. (We shall assume that we have only made two selections). 

Selection 1: 1 2 3 4 5 6
Selection 2: 1 2 3 4 5 7

So selection 1 and 2 have in common the numbers (1,2,3,4,5). We again have 408 winning combinations but this time Selection 1 has winning combinations in common with Selection 2. These are 38 combinations in common as obtained from my calculations. These 38 in common combinations are lost opportunities for a division 5 win. Let see if I can make this a little clearer. There are 204 ways in which Selection 1 can provide us with a Division Five win. The same applies for Selection 2. This makes 408 ways in which either Selection 1 or Selection 2 can provide us with a Division Five win. However certain numbers coming in will cause both Selection 1 and Selection 2 to win simultaneously. For instance if 1,2,3,4,5,12 came out of the barrel both our selections would win. These simultaneous wins provide us with multiple wins but does not help our chances in winning in the first place. In fact for each simultaneous win, of which there are 38 in number, must be subracted from our total of 408 to give us 408-38 = 370 possible chances for a win, in which 38 of these produce simultaneous or multiple wins. That is only 370 possible chances of winning with duplicates.

Duplicates therefore increase your chances of multiple wins but reduce your chances of winning at all. If your object is to increase your chances of winning, in the first instance, then minimize duplicates.

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