I will start by saying this isn’t a game changer either way. It’s a minor statistical anomaly that I’ve been thinking about lately, and yes, I do have way too much time on my hands.
Let’s say your hole cards are both spades, and 2 more spades come on the flop. Conventional thinking goes something like this…
There are 13 spades, I know the location of 4 of them. This means 9 spades are still out. A deck contains 52 cards, and I can see the 3 in the flop, plus my 2 hole cards. This leaves 47 cards unaccounted for.
Since there are 47 cards remaining and 9 of them are spades, the next card will be a spade 19% of the time. ((9/47)X100)
But there’s another way to look at it…
There are 9 people at this table, so 18 hole cards have been dealt. Of these, 4.5 will be spades, on average. I have 2 spades, so 2.5 spades additional are in the initial deal. The flop contained 2 spades, so I know the location of 4 of them, and statistically, 2.5 more are in other people’s hole cards. This leaves 6.5 unaccounted for.
I know my hole cards and the 3 in the flop. Although I can’t see the other 16 hole cards, I can discount them because statistically, they will contain 2.5 spades. Thus there are 31 cards unaccounted for, and 6.5 of these are spades. The odds of the next card being a spade is 21%.
The difference is only 2%, but which method is more correct, and why?
Sklanski’s fundamental theorem of poker basically states, “If seeing your opponent’s hole cards would make you play differently, your initial play would be incorrect.”
If you knew for a fact that 2 or 3 of your flush cards were missing from the remaining deck, would you still take that flush draw? Does Sklanski’s theorem even apply here?
I dunno. What do you think?