math question

If a player wins 1 time out 3 against an Ace, the player is a 34% underdog. With a 6 up-card, the player will win 8 times out of 13 or the player has about a 23% advantage. In real numbers what does the 34% and/or 23% ACTUALLY mean? Does it refer to the numbers of HANDS won or LOST or does it refer to the % of $$ won or lost or BOTH? I have thought about this a lot and I am not clear about the answer. A math persson (which I am not) will see this as a simple question. Thank you.

Quote: bobluered

If a player wins 1 time out 3 against an Ace, the player is a 34% underdog. With a 6 up-card, the player will win 8 times out of 13 or the player has about a 23% advantage. In real numbers what does the 34% and/or 23% ACTUALLY mean? Does it refer to the numbers of HANDS won or LOST or does it refer to the % of $$ won or lost or BOTH? I have thought about this a lot and I am not clear about the answer. A math persson (which I am not) will see this as a simple question. Thank you.

The numbers you are referring are called Expected Value (EV) or advantage.
The first number would be EV of -34% and the second EV of +23%.
It refers to amount of money won, not to amount of hands won ( if the only possible outcomes are win 1 bet ot lose 1 bet, then it would aslo equal to amount of hands won).
So for every $100 you bet and and Ace comes you expect to lose $34. So if you play 1 million hands of $100 where an Ace comes you expect to lose around $34 million.

I assume that the number you give are as an example and do not represent the actual probabilities.

But you are making an oversimplification with your example. Apart from Win 1 or Lose 1, another very common occurence in BJ is a Tie.
And also you have doubles and splits where you can win or lose 2 units and in some games surrender where you can lose 0.5 unit.

So the probabilities could be (just a random example)
Prob EV
Win 2 8% 16%
Win 1 40% 40%
Tie 0 10% 0%
Lose -1 37% -37%
Lose -2 5% -10%

Total 100% 9%
So in this example even though the probability of Win is less than 50% at 48% (8%+40%) there is a positive EV of 9%.

So in BJ even though the player wins quite a bit less often than the dealer (I think around 5% less) the EV is only around 0.5% because of BJ paying 3:2 and doubles and splits.

In a standard game, the possible outcomes from an Initial Hand can be:-8,-6,-5,-4,-3,-2,-1,0,1,1.5,2,3,4,5,6,8.
The +8 can come from splitting into 4 hands, Doubling all and winning all. A very rare case indeed which I do not think I have ever seen.