No the odds do not approach zero as the number of tickets sold approaches the odds.
Think of it this way: Suppose you had 100 balls floating randomly (as in a lottery machine by air flow) suspended above 100 cups. You turn the air flow off and the balls drop. Some of the balls will fall into cups in which other balls have already fallen. Some cups will have 2 balls, some will have more than two balls, and some will be empty. This is very much the same case as the lottery, except one is envisioning hundreds of millions of balls falling into hundreds of millions of cups in the case of MM and PB.
The type of mathematics that allows one to calculate the distribution of these balls is called a Poisson distribution.
Irrespective of the odds of a particular lottery, any time the number of tickets sold is equal to the odds of the lottery, there is a 36.8% chance that the lottery will rollover. Thus in the lottery you mention, 13.9 million in sales would have a 36.8% chance of rolling over, on average.
In the case of MM, it takes about 120 M in sales before the probability of the lottery rolling over falls below 50%. However in repeated trials, even an event with a probability of 10% will occur. This accounts for why we almost always see a winner before sales actually reach $120M on a single draw.
On a drawing with 300M in sales on a single draw, there would still be a better than 18% chance of a rollover. There would be about a 2% chance of 5 winning tickets though, and a 31% chance of one winner.
