Here are the probabilities of hitting the jackpot on lotto variants if assuming you place a bet on just one combination
6/42 = 1/nCr = 42!/(6!(42-6)!) = 1/5,245,786
6/45 = 1/nCr = 45!/(6!(45-6)!) = 1/8,145,060
6/49 = 1/nCr = 49!/(6!(49-6)!) = 1/13,983,816
About
-betting on lucky pick(randomly generated numbers) in every game?
Your combination has the same chances as every other combination out there. But i think if your combination is picked randomly, there is a greater chance that you are the only winner for the jackpot. Why i say that is cause people cant truly pick a random number no matter how we try, we have biases and we tend to go for numbers that is based on bdays and whatever. We also tend to spread that numbers out, but in a truly random number some of it will have consecutive numbers.
So if ever you are betting on lotto, i suggest change one number to greater than 31 and change another number and have it be consecutive to one of the digits youve picked. it will give you much better chance of being a lone winner of the jackpot, if ever your combination is picked.