A pair vs Blackjack

[timebomb]

Hi, everyone,

I was discussing probability with my daughter the other day and we were wondering if our math is correct. This is the question:

Imagine you are in a game of Blackjack. The dealer deals you 2 cards. Which is more likely? That the 2 cards are a pair or that the 2 cards turn out to be Blackjack?

Here's how we did the math:

The probability of 2 cards turning out to be a pair (any pair) - I ignore the first card. It does not feature into the calculation because the key lies in the 2nd card which has to be the same as the first. Let's say the first card is an Ace. It would mean there are only 3 aces left in the deck. There are 52 cards in a deck but since 1 card has been taken out, there's only 51 cards left. So the probability of a pair is 3 out of 51. Or 1 in 17.

The probability of 2 cards turning out to be Blackjack - One of the cards has to be an ace and the other either a 10, Jack, Queen or King. The probability of the first card being an ace is 4 out of 52. Or 1 in 13. The probability of the second card being either a 10, Jack, Queen or King is 16 out of 51. Multiply 1/13 by 16/51 and the answer is about 1 in 41.

If our math is right, it would mean the chances of getting a pair from 2 cards dealt randomly is higher than Blackjack - about 2.4 times (41 divided by 17).

Is this correct?

Please forgive my poor English; it's my second language.

[tot]

Forget it, I wasn't thinking...

[Modified]

The probability of 2 cards turning out to be a pair (any pair) - I ignore the first card. It does not feature into the calculation because the key lies in the 2nd card which has to be the same as the first. Let's say the first card is an Ace. It would mean there are only 3 aces left in the deck. There are 52 cards in a deck but since 1 card has been taken out, there's only 51 cards left. So the probability of a pair is 3 out of 51. Or 1 in 17.

Correct.

The probability of 2 cards turning out to be Blackjack - One of the cards has to be an ace and the other either a 10, Jack, Queen or King. The probability of the first card being an ace is 4 out of 52. Or 1 in 13. The probability of the second card being either a 10, Jack, Queen or King is 16 out of 51. Multiply 1/13 by 16/51 and the answer is about 1 in 41.Half right. The first card could also be a ten or face card and the second one an ace. So just double your answer.

Of course, single-deck blackjack in a casino is rare, because it's too easy to count cards.

[Rodney]

Half right. The first card could also be a ten or face card and the second one an ace. So just double your answer.
Right, which brings down the probability of blackjack to 1 in 20.7 (4.8% chance), not that much different than the 1 in 17 (5.9%) probability of a pair.

[maggs]

Correct.

Half right. The first card could also be a ten or face card and the second one an ace. So just double your answer.

Of course, single-deck blackjack in a casino is rare, because it's too easy to count cards.

Just for the sake of getting off topic I want to correct this single deck misconception.
The real reason you don't find many single deck games is because hands dealt per hour is low compared to a six or eight deck shoe games. If a Blackjack game has six players then the dealer will generally shuffle after only two rounds of play where as an eight deck game is shuffled only four to five times an hour. Also dealing from a shoe is faster than hand pitching. More hands per hour = more money for the house.
All games (single or double deck, four,six or eight deck shoe) are easy to count and the shoe is actually preferred by counting teams.

[Ladewig]

Just for the sake of getting off topic I want to correct this single deck misconception.
The real reason you don't find many single deck games is because hands dealt per hour is low compared to a six or eight deck shoe games. If a Blackjack game has six players then the dealer will generally shuffle after only two rounds of play where as an eight deck game is shuffled only four to five times an hour.

But I've seen single-deck dealers do a couple of riffle shuffles and and start dealing pretty fast. I'd like to actually time them and see if they can shuffle a single deck in one sixth the time it takes to shuffle six decks.

Also dealing from a shoe is faster than hand pitching. More hands per hour = more money for the house.

Again I think the difference in dealing time is close enough that I'd want to see them both timed with a stopwatch.

All games (single or double deck, four,six or eight deck shoe) are easy to count and the shoe is actually preferred by counting teams.

While eight decks with shallow penetration is countable, it is not often profitable.

In any case, many casinos are going to shuffling machines so that there is always a fresh set of decks available and others are going to continuously shuffled decks.

[maggs]

Ladewig you make some good points but let me clarify.

Overall shuffle times between a shoe and single deck would be similar but most casinos use the automatic shufflers on shoe games but not single deck (though some do) so getting the next shoe in action is very fast.

Some dealers can pitch cards about as fast as dealing out of a shoe (some can't even come close) but the big difference is the shoe game is dealt face up and the dealer can move around the layout very fast. Single deck pitch games are face down, the players handle their own cards and this is much slower. You would be surprised (maybe you wouldn't) how many full grown adults have trouble counting to 21 without the dealers help. If the dealer can't see your cards then he/she can't help.

If your not familiar with card counting teams they work like this: One player will sit down at a shoe game and count cards. When the shoe reaches a very favorable count he signals his team to jump in. They will play big money and stay until the counter tells them the count has gone bad. The team will have several tables being counted down throughout the casino. The team only plays hands that are favorable unlike the lone counter that has to play all hands though he bets less on the unfavorable ones.

You don't have the option of just jumping in on a single deck game and even if you did you would likely only get in one round before a shuffle and the count starts over.

You probably guessed by now that I am a dealer:(

[timebomb]

Thanks for the replies, everyone. I'm delighted there's a dealer in the forum :)

I won't describe myself as a gambler but I enjoy having a flutter with a game of cards every now and then. Here in Singapore where I live, the nearest casino and the only one around the region is the one in Kuala Lumpur, Malaysia known as Gentings Casino. Maggs, it may interest you to know that in Gentings, the house does not pay more for Blackjack. It's just even money. You would imagine that if it's just even-money for a Blackjack, there won't be any need for the casino to take counter-measures against card-counting. But no.

Gentings uses 8 decks of cards. Also, they don't deal to the last card but will stop and re-shuffle when there's about a deck left in the shoe. On top of that, they go through a certain ritual during every re-shuffle where a certain number of cards is discarded before play begins. These cards are discarded face-down and the number of cards to be discarded is determined by opening the first card in the shoe. For instance, if the first card is a Jack, they will throw 11 cards.

In such a situation, would I be correct to say that card-counting won't give the player any advantage?

[Modified]

All games (single or double deck, four,six or eight deck shoe) are easy to count and the shoe is actually preferred by counting teams.

I would think that with a single deck it would be a lot easier to track the entire deck and use that information to modify hit/stand/split/double decisions, rather than just tracking high/low and aces for bet changes.

[Modified]

Gentings uses 8 decks of cards. Also, they don't deal to the last card but will stop and re-shuffle when there's about a deck left in the shoe. On top of that, they go through a certain ritual during every re-shuffle where a certain number of cards is discarded before play begins. These cards are discarded face-down and the number of cards to be discarded is determined by opening the first card in the shoe. For instance, if the first card is a Jack, they will throw 11 cards.

It makes no difference whether they leave those cards out at the beginning or the end. Maybe this is to prevent some form of cheating other than card counting.

In such a situation, would I be correct to say that card-counting won't give the player any advantage?A player advantage may not be possible, but you can certainly decrease the disadvantage by counting.

[XBoxWarrior]

Talk about a good thread de-rail.........

Get on with the card counting!

(pairs vs. BJ) was dead in the water!

[The Pro from Dover]

What hasn't been taken into consideration is that when you get your second card, the dealer has also been dealt a card, so there are only 50 cards in the deck and the dealer may or may not have one of the cards you need. I'll leave the math for others who understand how to determine probabilities better than myself.

[LordoftheLeftHand]

Gentings uses 8 decks of cards. Also, they don't deal to the last card but will stop and re-shuffle when there's about a deck left in the shoe. On top of that, they go through a certain ritual during every re-shuffle where a certain number of cards is discarded before play begins. These cards are discarded face-down and the number of cards to be discarded is determined by opening the first card in the shoe. For instance, if the first card is a Jack, they will throw 11 cards.

Since the cards that are set aside are not shown to the players they have no change on the game. They are identical in all respects to the last bunch of cards that are never dealt from the shoe.

I think the biggest change is the paying less for a "blackjack". I don't know if this would make it officially unprofitable, but it certainly would make it less profitable.

LLH

[Ladewig]

.

Some dealers can pitch cards about as fast as dealing out of a shoe (some can't even come close) but the big difference is the shoe game is dealt face up and the dealer can move around the layout very fast. Single deck pitch games are face down, the players handle their own cards and this is much slower.

I forgot about that. You are right.

[Ladewig]

What hasn't been taken into consideration is that when you get your second card, the dealer has also been dealt a card, so there are only 50 cards in the deck and the dealer may or may not have one of the cards you need. I'll leave the math for others who understand how to determine probabilities better than myself.

I know only a little probability, so perhaps I am missing something, but I am pretty sure that the dealer having a face down card doesn't change the probabilities or the strategies based on those probabilities.

[sol invictus]

I would think that with a single deck it would be a lot easier to track the entire deck and use that information to modify hit/stand/split/double decisions, rather than just tracking high/low and aces for bet changes.

It is.

The biggest obstacle to card-counting is the pit boss. The thing is, you have the most information when most of the deck has been dealt. Maybe 1/4 of the time the count will be pretty favorable. So, if you could play with the optimal strategy, you'd bet the minimum most of the time, and then roughly 1/4 times when the deck gets low you'd raise your bet to the maximum. But that pattern is really obvious to anyone keeping track, and so (at least in Nevada) you can't play that way.

[Ladewig]

Maggs, it may interest you to know that in Gentings, the house does not pay more for Blackjack. It's just even money. You would imagine that if it's just even-money for a Blackjack, there won't be any need for the casino to take counter-measures against card-counting. But no.
that


This site (http://wizardofodds.com/blackjack) says that paying 1:1 for blackjack adds 2.27% to the house advantage. I think that is larger than the edge gained by most counting systems, but I will defer to the pros and semi-pros around here.