Video Poker Odds With Replacement

For reasons that I won’t go into right now, I was curious about the odds of drawing different types of poker hands if you draw cards with replacement (i.e. put each card back after drawing it). The most obvious difference is that this opens up some new possible hand types, like five-of-a-kind and even five-of-a-kind flush, but it’s also interesting to see how drawing with replacement changes the odds of getting certain kinds of hands.

Wikipedia has a nice breakdown of the probability of different poker hands when drawn the normal way without replacement, where they compute the number of hands (that is, unique sets of 5 cards where order doesn’t matter) and then divide by $\binom{52}{5}$ = 2,598,960. Turns out when computing the probability with replacement it’s easier to instead count the number of card sequences (including order) that match a given hand type and then divide by 52⁵. Here’s the chart, along with the formulae:

Hand	Formula	Frequency	Probability
Royal Flush	$\binom{4}{1} 5!$	480	0.000126%
Straight Flush	$\binom{4}{1} \binom{9}{1} 5!$	4320	0.001136%
Five of a Kind	$\binom{5}{5} \binom{13}{1} 4^5$	13,312	0.003501%
Four of a Kind (excluding 5K)	$\binom{5}{1} \binom{13}{1} \binom{12}{1} 4^5$	798,720	0.2101%
Full House	$\binom{5}{3} \binom{13}{1} \binom{12}{1} 4^5$	1,597,440	0.4202%
Flush (excluding above)	$4 [13^5 – \binom{10}{1} 5! – \binom{13}{1} – \\ \binom{5}{1} \binom{13}{1} \binom{12}{1} – \binom{5}{3} \binom{13}{1} \binom{12}{1}]$	1,470,960	0.3869%
Straight (excluding above)	$\binom{10}{1} 5! (4^5 – 4)$	1,224,000	0.3219%
Three of a Kind (excluding above)	$\binom{5}{3} \binom{13}{1} \binom{12}{1} \binom{11}{1} (4^5 – 4)$	17,503,200	4.6036%
Two Pair (excluding above)	$\binom{5}{1} \binom{13}{1} \binom{4}{2} \binom{12}{1} \binom{11}{1} (4^5 – 4)$	26,254,800	6.9054%
Jack+ Pair (excluding above)	$\binom{5}{2} \binom{4}{1} \binom{12}{1} \binom{11}{1} \binom{10}{1} (4^5 – 4)$	53,856,000	14.1650%
Low Pair (excluding above)	$\binom{5}{2} \binom{9}{1} \binom{12}{1} \binom{11}{1} \binom{10}{1} (4^5 – 4)$	121,176,000	31.8713%
No Hand	$52^5$ – all above	156,304,800	41.1108%
Total	$52^5$	380,204,032	100%

Compare to the normal draw without replacement probabilities. In regular poker the chances of getting n-of-a-kind decreases dramatically as n increases because with each instance of a rank you already have that’s one less in the deck you can pull. Not so when you’re drawing with replacement, which bumps the chances of getting n-of-a-kind dramatically. On the other hand, the chances of getting a straight and its variants is actually helped by drawing without replacement, because every card you’ve drawn so far is one less card you could draw to blow your straight, so the probability of one of those variants goes down by 18%.

Hand	Probability with replacement	Probability without replacement	Change
Royal Flush	0.000154%	0.000126%	-18%
Straight Flush	0.00139%	0.001136%	-18%
Five of a Kind	0%	0.003501%
Four of a Kind	0.02401%	0.2101%	+775%
Full House	0.1441%	0.4202%	+192%
Flush	0.1965%	0.3869%	+97%
Straight	0.3925%	0.3219%	-18%
Three of a Kind	2.1128%	4.6036%	+118%
Two Pair	4.7539%	6.9054%	+45%
Jack+ Pair	13.0021%	14.1650%	+9%
Low Pair	29.2548%	31.8713%	+9%
No Hand	50.1177%	41.1108%	-17%
Total	100%	100%