Blackjack math - what am I missing?

Hi all. Can someone tell me if and how my math is wrong or my logic is wrong?
Generally speaking, with differences based on rules, decks, etc., you will experience these hand results;

player win 42.5%
player lose 49.0%
push 8.5%
player blackjack 4.75%

So, if you play 100 hands you theoretically should win 43ish, lose 49ish, and push 8ish. Also, 5 of the winning hands should be blackjacks.

If you put dollars to this, playing $10/hand;

win - 38 x $10 = $380
win (with BJ) - 5 x $15 = $75
lose - 49 x $10 = ($490)
push - 8 x $0 = $0
Total = 100 hands = ($35) loss

If the theoretical loss for playing 100 hands at $10/hand is $5 ($10 * $100 * .005), where am I making up the difference to my example above? Should I assume the difference is made up in double down hands? If so, how often are double down opportunities dealt, and how often should they win?

Appreciate the help in advance and apologize if this has been covered before.

Vinny,

As you realized, you failed to account for double downs as well as splits.

For a double after split game, on average in 100 rounds you will play about 102 hands.

Clearly, all DD hands are +EV (else why would you double?) as are most splits. The added EV from these hands nearly compensates for the dealer's advantage of playing second, which is why the house edge is so small.

Hope this helps!

Dog Hand

My stats on the home game have a different slant.
Out of 46,496 hands played, 37,601 were Normal hands (80.87%); 3,761 hands were hard double downs (8.089%); 750 hands were soft double downs (1.613%); 23 were insured hands (not supposed to insure hands but I insured them anyway once in awhile and very rarely, so minus half a bet or even money on a BJ) 23 hands 0.05%; 2,041 BJ's (pays 3:2) 4.39%; Surrender (lose half a bet) 2,343 hands 5.04%.

On another chart I get results of: 40.79% Normal hands won; 57.43% of hard double downs won; 54.40% of soft double downs won; 21.74% of Insurance bets won; 100% of BJ's won (but it doesn't differentiate between 3:2 wins and even money payouts); and I lost all my surrender bets. I won 42.90% of all bets including pushes.

But don't get hung up on those averages, they're just averages. In the short run where you get streaks and runs and sessions, your results could veer far from the average.

DogHand, thank you that does help.
It makes sense that approximately 2 split opportunities will arise for every 100 hands dealt.

I also went ahead and made the effort to calculate that there should be a double down opportunity on about 10% of hands dealt. So 10 hands in my example.
So to answer my own question; 10 out of 100 hands will be dd opportunities, and roughly estimating, that would result in 5 wins, 4 loss, and 1 push, which would make up for most of the difference.

Does that sound right?

Thanks ChumpChange.
I understand that the true averages and probabilities play out over thousands of hands. I am trying to understand the probabilities more precisely so that when things are trending (usually losses) I can gauge just how far off it is from expected.

If I ever get around to playing at a table and score a big win, I should note that the end of a shoe could signal the turn of my good luck to bad luck. So if I already hit my win goal for the day or the session, that's a good cut-off point.

.
if you decide to learn to count where as I'm sure you know the player can gain an advantage then I recommend:

"Professional Blackjack"

by Stanford Wong

the Wizard also has an excellent summary of this - linked


https://wizardofodds.com/games/blackjack/card-counting/introduction/


.

I've toyed with the idea of counting but determined that I don't have the time, bankroll, or attention span that is required to do it successfully.
My question developed after playing a few thousand hands online. I've played through some wicked streaks which left me questioning how "normal" the play was and/or how legit the software was.
In any case I enjoy the math as well.