Playing with an infinite copies of a deck

Question

Let's assume we are no longer having a deck with 52 cards, but infinite cards, where every card has a probability of 1/52 to be received. With the logic, if you have two cards, the probability for a combination is no longer 1/52 * 1/51, but (1/52)^2 (that's the condition of the infinite deck).

Question: Will the order of hands be swapped? What will be the probabilities (and probabilities difference) for each hand? Which will be more often, which less?

Answer 1

The new rule makes the probability of each card 1/52. But you will still have the same behavior in the "hand rareness" sense of the game.

making one pair is easy because you only need 2 cards of the same number (each number has 4 occurrences in the same 52 card-set). So it will be (52/52)x(3/52)x(52/52)x(52/52)x(52/52) = (3/52).

3/52 only because I assume the first card was drawn in the first slot and there are 3 occurrences left. If you say drawing a card doesnt actually eliminate that card from the probability, then the probability is 4/52.

However, a royal flush is way harder because you will need 1 specific card in each slot of the 5 cards hand. So it will be (1/52)x(1/52)x(1/52)x(1/52)x(1/52) = (1/52)^5

Answer 2

Was hoping someone would do the math but what would happen is the convergence of rareness for 5-card hands. If the value of a hand is based on its improbability (which is not always the case), then the values of 5-card hands also converges.

The way i look at it is outs. In particular, hands which block their own outs increase in probability, while hands that don't, decrease in probability. For eg, a pocket pair is now 33% more likely than before, because after drawing your first card, you have 4 outs to make a pair, instead of 3. You are twice as likely to flop a set (4 outs instead of 2), and twice as likely to hit a boat/quads by the river. Hence, the value of boats/quads is significantly lower.

For a flush, its somewhat similar, but the difference in outs is less significant. For eg, a flush draw is now drawing to 13 outs instead of 9. Not double the probability like the previous scenario, but still about 45% more likely.

For a straight, the reverse is true. Straights dont block their own outs, so a OESD would still have 8 outs. Since you are now drawing from a 52 card deck instead of 47/48 card deck for turn/river, your odds actually decrease. Thus, the value of a straight should increase.

You also have to have extended rules for new hands, i.e. 5 of a kind, exact pairs, 5 of a kind flush?. Also, a royal flush is almost equally unlikely (kinda disagree with jackhammer that its way harder).

Answer 3

I happened to search for the same question myself and found this list of theoretical hand rankings. Unfortunately the author doesn't show their working or detail what the priorities might be, nor do they choose to artificially elevate an ace-high straight flush in importance above other straight flushes, but it is nevertheless an (at least plausible) answer to the question, and I do like the silly names suggested for the potential new hand types that the infinitely large deck enables.

Answer 4

I have a program that simply enumerates every possible 5-card hard and displays the odds. It's pretty easy to modify this for combinations with replacement and see the difference. For the normal 52 card deck, you get the standard odds and hand rankings:

hand type          count    1 in x odds    percentage
---------------  -------  -------------  ------------
straight flush        40          64974        0.0015
four of a kind       624           4165        0.024
full house          3744            694        0.1441
flush               5108            508        0.1965
straight           10200            254        0.3925
three of a kind    54912             47        2.1128
two pair          123552             21        4.7539
one pair         1098240              2       42.2569
all hands        2598960              1

But changing it to allow replacements with equal probability:

hand type          count    1 in x odds    percentage
---------------  -------  -------------  ------------
straight flush        40          95495        0.001
five of a kind       728           5247        0.0191
straight           10200            374        0.267
four of a kind     21840            174        0.5718
flush              24712            154        0.6469
full house         31200            122        0.8168
three of a kind   274560             13        7.1878
two pair          343200             11        8.9847
one pair         1830400              2       47.9185
all hands        3819816              1

Straights (and straight flushes) are harder to make since having duplicate cards don't really help.

All other hands are easier to make, but the full house and four-of-a-kind are the big movers. Full house becomes easier to make than a flush or a straight.