This might be trivial, and then again, it might not be.

Two guys at my home game started a prop bet on the sum of the flop cards. They set the over/under at 21 and each took a side (this came out of a discussion on blackjack, hence the 21).

All face cards are worth 10, aces are 11, and the numbered cards are worth their face value.

For example, a 4 9 K flop equals 23 and the over wins. All 21's are a push.

Who has the edge? What's the expected value of the flop sum?

Some simple match revealed the average value of one card is approximately 7.3, therefore three cards would be 21.9. It *can't* be that basic, right?

This seems like a combinatorics problem, but I don't have the math skills to work it out easily.

For what it's worth, the guy with the under played it the right way. They both lost track of the prop after a while and he made a point of only reminding the "over" player when the under hit.

actually it is. this is how distribution of the sum looks like.

mean is 21.9231.

total number of flops is 22100. number of strictly lower flops than 21 is 8292, so the chance for under is 37.52%.

Now you just need to know the chance of exactly 21, from which you can deduce P(over) and then the edge.

less than 21: 8292 (37.52%)
exactly 21: 1508 (6.82%)
more than 21: 12300 (55.66%)
total: 22100

i thought OP was just curious which was more likely. and he didn't say the odds for the bet, I guess 50-50? if so, the one who went with over would have 18.14% return on his stake, while the guy with under is losing at the same rate.