Card values for different payouts on naturals

[Lonesome Gambler]

I'm looking (unsuccessfully) for the value of each initial card in games where the payout on a natural is something other than 3:2.

For example, what is the EV of receiving a Ten as your first card in a 6:5 game? Obviously, receiving an Ace will be far less valuable in a 6:5 game than in a 3:2 game, but what's it worth in a 2:1 game? How about in an even-money game? For simplicity's sake, let's forget about the 7:5 game...

I'm assuming that the EV of a card in the dealer's hand won't change, since the dealer always gets even money on a natural anyway. But what about in games where the dealer pushes on 22, etc.?

Summary: does anyone currently have the initial card values for different payout games (6:5, 1:1, 2:1), for both cards landing in the player's hand and in the dealer's hand? And if this is a lot to ask, can anyone help me figure out how to calculate this information on my own?

PMs welcome. Thanks!

 

[MangoJ]

Thats simple to answer. If X is the EV of the first card in a 3:2 game,
then
X = pBJ * 1.5 + (1-pBJ) * EV(non-BJ)

where pBJ is the probability of a "paid" Blackjack (that means, excluding a BJ push).

Now if you ask for Y, the EV of the first card in a 6:5 game, it is:

Y = pBJ * 1.2 + (1-pBJ) * EV(non-BJ)

If you know X and pBJ, you can solve for Y:

Y = X - pBJ * (1.5 - 1.2)

For your first card Ace:
X = 50.4%
pBJ = 4/13 * (1-8/169):
EV(Ace) = 41.6% (6:5), 35.7% (1:1), 65.1% (2:1)

For your first card Ten:
X = 14.3%
pBJ = 1/13 * (1-8/169):
EV(Ten) = 12.1% (6:5), 10.6% (1:1), 18.0% (2:1)

All for S17 games according to WoO

I'm assuming that the EV of a card in the dealer's hand won't change, since the dealer always gets even money on a natural anyway.

It is right, the dealer always collect same amount on BJ, but he has more "at risk" (potential payout) in a 2:1 game compared to a 1:1 game. This reduces the house-EV for every dealer upcard (not only Ten and Ace) - as can be easily seen from the players view. If you need help for dealer cards, PM me.

 

[Lonesome Gambler]

Excellent, thank you!