The lottery in our state consists of two drawings. First, a MegaBall is picked from among 27 numbered b***s. Second, five WinnerBalls are pi

You are right Mellie.  I did this one too quickly (as I often do!).  I've now corrected it.

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Probability of getting Megaball:  1/27

Probability of getting 5 Winnerballs = (5/44)*(4/43)*(3/42)*(2/41)*(1/40)

Overall probability of winning = (1/27)*(5/44)*(4/43)*(3/42)*(2/41)*(1/40) = 1/29322216 ≈ 3.4*10^-8

Edited to correct error

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$$The probability that the MegaBall matches is $\dfrac{1}{27}$ . The probability that the 5 WinnerBalls match is $\dfrac{1}{\binom{44}{5}}$. So my chances of winning are $\left(\dfrac{1}{27}\right)\times\left(\dfrac{1}{\binom{44}{5}}\right) = \boxed{\dfrac{1}{29,\!322,\!216}}$.$$