Answer
118 million.
Work Step by Step
We must have $E(prize)\gt1$
$E(prize)\gt1$
$(grand~prize)\times0.00000000684+200,000\times0.00000028+10,000\times0.000001711+100\times0.000153996+7\times0.004778961+4\times0.007881463+3\times0.01450116+0\times0.9726824222\gt1$
$(grand~prize)\times0.00000000684+0.197\gt1$
$(grand~prize)\times0.00000000684\gt1-0.197$
$(grand~prize)\times0.00000000684\gt0.803$
$(grand~prize)\gt\frac{0.803}{0.00000000684}$
$(grand~prize)\gt\frac{0.803}{0.00000000684}$
$(grand~prize)\gt117,397,660.82$
That is, the grand prize must be at least 118 million so that $E(profit)\gt0$
