If the lottery is truly random, then it shouldn't make any difference if you choose the same number or not.

The biggest difference in a lottery is that you are best off choosing a number that nobody else has chosen. Why share a pot 5 ways when you can get the whole thing yourself?

Thus, if a bunch of people restrict their lottery choices to numbers between 1-12 & 1-31, then they may have a greater probability of sharing the pot, and thus receiving a smaller pot.

As I calculated above, if you always guess red, you'll have an overall 80% chance of being right.

If you guess randomly 50/50, then you'll have an overall 50% chance of being right,

And if you guess 80% of the time red, and 20% of the time green, you'll have an overall 68% chance of being right.

The only difference is if the trials aren't independent. Then your odds may change slightly. This what some people get with "counting cards" when playing blackjack. Simply considering the 10,J,Q,K, or 16 cards out of 52 cards, one gets a probability that these cards will come up (4:13). However, based on the history of a deck, the probability of those cards can change significantly. Likewise, if you're wishing for an ace, the probability on a full deck would be 1:13, but if half the deck was played without any aces showing up, then the new probability is 1:6.5, or much better odds.