"There is always a risk of several self pick players picking the same five numbers, but at best it's only a small percentage of all ticket sales."
That "small percentage" is actually about 20 to 30%, and as I said in my first post, many of those players don't choose their numbers randomly. That means there are more duplicate combiantions from that group than their percentage would suggest. Because birthday numbers account for less than 5% of possible combinations, selling 70 to 80 random QP's should result in only 3 or 4 tickets with those combinations. 20 to 30 self pick tickets would be expected to use 3 or 4 birthday combinations if just 10 to 20% of players use all birthday numbers when making their selections. Players choosing their own numbers very definitely results in far more duplication than would result from purely random selection.
The ultimate example of the lottery having an unusually large number of duplicate combinations is the fortune cookie powerball drawing from several years ago, when there were more than 100 5+0 winners. We can also be sure from tha incident that there were many more players who played other sets of numbers from fortune cookies, so there were probably hundreds of combinations that were played 50 times or more.
Many years ago the NY lottery ran advertisements that said that nearly 10,000 thousands of people played each of the two combinations made by choosing the numbers diagonally from the top left and top right of the bet slip. The lotteries have also reported that large numbers of people play other patterns from the bet slips, consecutive numbers, and multiples of 5.
That makes the chance of an unusually high number of duplicates from randomly selected numbers completely insignificant compared to the risk of letting people choose ther own numbers.
Of course none of that means the lotteries are actually taking a risk. First, everything that results in an increased risk of multiple winners for some combinations is balanced by a reduced chance of having even one winner for other combinations. If drawing all birthday numbers increases the chance of Rhode Islnd having two 5+0 winners out of only 56,000 players, it also reduces thier risk of paying even one prize when the winning combination is from the more than 95% that have a non-birthday number. In the long run, probability ensures that if Rhode Island pays two winners in one drawing it will be balanced by an unusually long run with no winners.
Second, just in case probability doesn't protect them, the official rules will. All lotteries have arule that allows them to make the prizes parimutuel if there are an unusually high number of winners. PB could have exercised that option when they had the fortune cookie payout, but chose to absorb the additional expense. If the winning combination ever is 1,2,3,4,5 or one of the other heavily played combinations you can be sure that the lotteries will exercise that option.
The lotteries have absolutely no need to deliberately reduce duplicates by preventing purely random selection of numbers. On the contrary, a purely random selection reduces their chance of having to reduce individual prizes by making parimutuel payouts.