What am I missing here? Why would you take 4-1 on Any 7 when you could take all three hop bets and get essentially 5-1 on it?

It's not exactly that. $3 bet hop any seven, hit leaves you holding $15. $1 each on every combo. of seven, hit leaves you holding $16 for the same $3. Little bit better, but not night and day. White Moses, that's good.

Clearly the hoppers are a better bet...providing you are betting in three-unit increments. Same situation with corner red (Big 6 and Big 8). Placing those numbers pays 7:6 but on a $5 table requires a $6 bet.

That one by her, was so bad, I decided to ignore it, which I'm thinking will induce a temper tantrum, which should be quite humorous.

The bet pays 15 to 1; it DOES NOT matter if you hop one 7 combination, 2 seven combinations OR all 3 seven combinations; each winning bet will pay $15 to $1. You are confusing $16 FOR $3 with 15 to 1.

Im not confusing anything. The only way to compare the two bets, TD, is is to have an equal amount bet, on the winning combinations. A winning "red" bet, is achieved with a roll of 6/1, 5/2, or 3/4. Which if played as a hop bet, would require three bets of one unit each. So, the fair comparison of the two bets would be, a three unit "big red" vs 3, one unit bets, on each of the hop combinations. That it would take to cover, any GD seven. A $3 "red" winner nets $12 A $3 "hop all 7s" winner nets $13. This example, is for random dice, if the dice are not random, I'll let you, cover all of the other possibilities.

If you bet Any 7 for $3 and one hits, you get paid $12 if the payout is 4-1 and the $3 stays up. You could hop each 7 for $1 and get paid $5 but you would lose the other two bets. Or, you could bet $3 on each ($9 total) to get $15 if the payout is 5-1, but you would lose the other two $3 bets.