Disclaimer:
#1 I don’t buy lottery tickets
#2 I have set up trusts for clients who have won multi-million dollar lottery prizes
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That said, when purchasing a ticket for a lottery that has a Shabbos drawing, you are NOT buying a chance for a prize that will definitely be awarded on Shabbos. Very often there is no winner and the pool grows and the next drawing occurs on a weekday.
This is different than buying a raffle ticket with a set prize drawing on Shabbos…knowing that all prizes will
be awarded then.
We had this discussion with our shul Rav when all houses of worship in our town were asked to sell raffle tickets to members to raise funds for a new ambulance for the volunteer corps. A Friday night was set for the drawing. Our Rav insisted that the drawing be changed to Monday so Jews,Christians and Moslems would not have issues with the date.