I recently received a letter from someone who asked if more quads started as a single pair or as three of a kind? An interesting question to examine with not as easy an answer as you might expect. And since the letter was referring to games without wild cards, I’ll limit this discussion to that as well.
Interestingly, a single pair is dealt exactly 20 times as often as trips are dealt. Not ABOUT 20 times. EXACTLY. I’m sure there is a simple explanation for this, but I have not yet discovered it. Drawing three cards, you will convert a pair into quads every 360 times (rounded). And drawing two cards, you convert trips into quads every 23.5 times. Since 360 is only a little more than 15 times as much as 23.5 and 15 is less than 20, it would seem that when the calculations are compared, quads start off as single pairs more often than they start off as trips.
But not so fast. It depends on the strategy used. For example, how do you play 5♥ 5♠ 6♠ 9♠ A♠? In most games, a 4-card flush is preferred to a low pair, so even though you are dealt a pair, and the hand is counted as being among the ones with starting pairs, you are not always going to draw to it. Or how about 4♦ 4♠ 5♥ 6♣ 7♦? In Jacks or Better, you should draw to the pair, but in Double Bonus, when straights pay 5-for-1, you should draw one card to the open-ended straight. Or how about A♥ A♣ 2♣ 4♣ 5♣? All beginners draw to the aces, but knowledgeable players in almost every game (including Double Bonus and Double Double Bonus) draw to the clubs and hope for straight flush.
I cannot think of any example where trips are dealt and wouldn’t draw to them, but there is one common example where you draw to trips when you weren’t dealt “trips!” This might sound impossible, but it’s not. In Double Bonus and other games where four aces pay a bunch, from a hand such as A♥ A♠ A♦ 4♣ 4♥, knowledgeable players toss the pair of fours and draw to the aces. The hand wasn’t originally counted as a trip, because it was counted as a full house — which is an entirely different category. So this is a case where a quad can start from “trips”, even though the hand wasn’t included in the “trip” count.
In a more volatile game, Triple Double Bonus, from A♥ A♠ A♦ 4♣ 4♥ you hold AAA4.
Many of the same games where you draw to trip aces even though you were not dealt “trips” also lead to plays where you draw to a pair of aces when you were dealt two pair. That is, if you were dealt A♥ A♣ 5♦ 5♠ 3♥, it is often correct to hold just the aces. But in other games, you should hold the two pair. (As a side note, it is almost never correct to just hold the kings from K♥ K♠ 5♦ 5♠ 3♥. It is usually MUCH better to draw to the two pair.)
So we cannot answer the question definitively as to whether quads start more often from trips or pairs unless we know what strategy you are using. Notice I said strategy, not game. The game only matters if you are playing the game correctly. For example, if you use the identical strategy on Jacks or Better, Double Bonus and Double Double Bonus, you will end up with the same hand frequencies at the end. Your score will be different in the various games because the games reward the same hands differently, but the hand frequencies will be the same.
But we can say that if you play correctly in most games without wild cards, ending quads are slightly more likely to have started from drawing three cards to a pair than from drawing two cards to trips. This is not because it’s easier to fill in a pair than trips (it’s not), but rather that you are dealt pair so much more often.
I find this interesting and something I had not known before I researched this in response to the letter.