The odds of winning are 1 in 302,000,000.
You get about 52% of the prize if you take the immediate option. You do have the option of getting the full amount, of course, if you take the annuity.
So you "only" need the prize to be $922m for the odds to be in your favor, which is nowhere near the $3 billion you're claiming. $922m x 52% if you take the cash is $479,440,000. Taxed 37% on that leaves you with $302,047,200. Better than the 1:302m odds. Of course if you live in a high-tax state that will change the numbers a bit, but in general yes the odds are technically in your favor when the prize is over $1b.
Actually do the math if you're going to pretend you've done the math.