Need help on math for new game

.....

If it's any help, when working out whether a card game works (most others are simple to work out) I start out using infinite deck analysis. This over estimates the chances of pairs (at 1/13 as opposed to a lower value) and flushes; but under estimates other chances (e.g. straights with say one deck: the chances of 6-7-8 in order are not 1/13 1/13 1/13 but 1/13 4/51 4/50). I also assume the player's cards don't affect the dealer's, in say a single deck poker game. This first approximation gives an impression of how good the game / payout schedule is and possible strategy.

For instance I was playing a game last week, I'm guessing Geoff had designed the payout for a flush so it was worth going-for-it (if Geoff sees this, it was XXpoker with three suited cards). One aspect of the game is you can increase your bet after seeing three (of six) cards towards a five-handed poker hand. The logic I used was (you get three common cards later) the chances of making a flush given any three suited cards are 1/4 1/4 1/4 + 1/4 1/4 3/4 + 1/4 3/4 1/4 + 3/4 1/4 1/4 = (1+3+3+3)/64 = 10/64 (pays 5 to 1) and you might make trips etc., so your expected return>0 - hence add-on.

Using similar reverse logic, when designing a game it's sometimes nice to give players some interesting decisions by setting the pay tables accordingly. Also I like it on a raise or fold type of game (with dealer qualifying or not), if the decision is close but not too terrible for people playing blind. It also gives options to play odds-on (play blind) or odds-against (look).

Quote: charliepatrick

If it's any help, when working out whether a card game works (most others are simple to work out) I start out using infinite deck analysis. This over estimates the chances of pairs (at 1/13 as opposed to a lower value) and flushes; but under estimates other chances (e.g. straights with say one deck: the chances of 6-7-8 in order are not 1/13 1/13 1/13 but 1/13 4/51 4/50). I also assume the player's cards don't affect the dealer's, in say a single deck poker game. This first approximation gives an impression of how good the game / payout schedule is and possible strategy.

For instance I was playing a game last week, I'm guessing Geoff had designed the payout for a flush so it was worth going-for-it (if Geoff sees this, it was XXpoker with three suited cards). One aspect of the game is you can increase your bet after seeing three (of six) cards towards a five-handed poker hand. The logic I used was (you get three common cards later) the chances of making a flush given any three suited cards are 1/4 1/4 1/4 + 1/4 1/4 3/4 + 1/4 3/4 1/4 + 3/4 1/4 1/4 = (1+3+3+3)/64 = 10/64 (pays 5 to 1) and you might make trips etc., so your expected return>0 - hence add-on.

Using similar reverse logic, when designing a game it's sometimes nice to give players some interesting decisions by setting the pay tables accordingly. Also I like it on a raise or fold type of game (with dealer qualifying or not), if the decision is close but not too terrible for people playing blind. It also gives options to play odds-on (play blind) or odds-against (look).

Really interesting insights there, charliepatrick. Thanks!

Quote: charliepatrick

I also assume the player's cards don't affect the dealer's, in say a single deck poker game.

You seem to imply that this assumption is some kind of approximation. As long as the "dealers strategy" (i.e. action of the dealer defined by the rules of the game) are independent of the players cards, this assumption is always true.

Quote: MangoJ

You seem to imply that this assumption is some kind of approximation.

As it happens with 3-card poker the decision whether to play or fold boundary is still Q64 whether you assume dealer's cards come from a separate deck or one using the remaining 49 cards.

Where there are no cards removed from the deck Q64 has expected return of -.995 475 (Q63 -1.006 335); where using removal of cards Q64 has expected return of -.993 378 (3 suits) / -.993 921 (Q6s) / -.994 138 (Q4s) / -.994 627 (64s). and Q63 > -1.002 551.

The player's expected values and strategy are based on what happens to the dealer after the player has made their decision; while the dealer's actions will be the same for any card combinations of the player (except taking even money, surrendering, busting etc. where no actions are needed) the probabilities of each possible outcome will be different. As a consequence this may affect a player's original strategy. Thus it is normal to hit 14 vs 10 in Blackjack, but in a single deck a player doesn't hit 7-7. The chances of player drawing a 7 for 21 is lower and the probabilities of dealer getting 10-7 is lower (a result which pays the player if they had drawn 4-5-6-7 and hence favour the player drawing).

Quote: charliepatrick

...One aspect of the game is you can increase your bet after seeing three (of six) cards towards a five-handed poker hand. The logic I used was (you get three common cards later) the chances of making a flush given any three suited cards are 1/4 1/4 1/4 + 1/4 1/4 3/4 + 1/4 3/4 1/4 + 3/4 1/4 1/4 = (1+3+3+3)/64 = 10/64 (pays 5 to 1) and you might make trips etc., so your expected return>0 - hence add-on.

Good work. You have come up with the correct play Charlie as it is correct to add on with any 3 suited cards.

.....