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The Law of Large Numbers, Splitting Tens and Doubling Down

I have a question about probabilities that has been bugging me since way back in college and I was hoping you could enlighten me.

There are two laws which, independently, I feel quite comfortable with; but which I can't seem to reconcile with each other.

(1) Law of large numbers (allow me to paraphrase) which states that after a million spins of a roulette wheel, the final results should be very close to 500,000 blacks and 500,000 reds (assuming for simplicity that there are no greens).

This is true.

(2) Law of independent events which states that the past has no effect on the future when you're dealing with independent events like dice rolling, roulette wheels, and, if I might add, blackjack games where the cards are shuffled after each hand.

Also true.

But I have great difficulty when I imagine a scenario where after 500,000 spins, black is ahead 275,000 to 225,000 (quite possible, wouldn't you agree?)

Not likely in that large a sample, assuming the wheel is in balance.

Wouldn't the odds on the next 500,000 spins be skewed towards red for the first rule to be correct? But obviously, this can't be right. Sample size isn't large enough, you might say. Okay, so let's make it 100,000,000 spins. After 50,000,000 spins isn't is possible for black to be ahead 27,500,000 to 22,500,000?

Again, not likely. Remember that as the sample size gets larger, the percentage of each event will get closer and closer to the expectation. If we were flipping a coin, the expectation is 50% 'heads' and 50% 'tails'. In a sample of 10 tosses, we'd expect 5 heads and 5 tails, but may well have 7 heads and 3 tails. However, as our sample size grows, we'll get closer and closer to 50%, BUT the actual number which one side leads the other will increase. For example, at 100,000 tosses we may be at 49.5% heads and 50.5% tails, thus tails lead heads by 1% or an actual number of 1000. At 1,000,000 tosses, we may be at 49.75% heads and 50.25% tails, but tails has actually come up 50,000 more times than heads!! At what point does it even up? That's the question you're asking and the answer is: It may never hit exactly 50-50. So, in a practical sense, you could spend a lifetime betting on the 'catch up' and never see it. Remember that my example is a 'fair' game and the house has a huge edge in roulette, so it's even more difficult to capitalize on this.

Let me start out by saying I think your column is fantastic. I've been practicing counting cards at home and will make my first attempt in a real casino this weekend with my buddy when he comes to visit (I'm doing an internship in Germany). I'll be playing with pennies for a while until I get the hang of counting, but your column really has me looking at the game from a completely different angle. Well done.

Quick question about one of your quiz questions. It states that when the player has a 6 and 3 and the dealer's face card is a 7, the correct play would be to hit. I would argue the correct play is to double, no? One would be a fool not to minimally hit when its mathematically impossible to bust, but that isn't my motivation. In all likelihood, you would double into a 19 and the dealer would end up with a 17. If he drew less than a 10, he would then be forced to hit again and very likely bust. If he drew an ace, he would end up with 18 and you would still win. So I would double in that situation...or is there some angle I'm missing?

No, the correct play is to hit. We don't assume you're going to get a 10, nor do we assume the dealer will end with 17. We analyze each play according to ALL the remaining cards in the deck and, based on that, the best play is to hit. If you hit, you can expect to win 17.4% of all the money you bet in that situation. If you double, you can expect to win just 11.4% of all the money bet. As a counter, when the true count gets to 6, THEN you would double 9 vs. 7.

I'm new to the basics of blackjack. But a main objective is to bust the dealer, and the best busting hand is a dealer showing 6. I also understand that a pair of tens is the second best hand in blackjack, but wouldn't it be an advantage to spilt tens to a dealer's 6?


Dear Rich,
First, the first object of the game isn't to 'bust' the dealer; the object of the game is to beat the dealer and that's done in a variety of ways. But splitting tens isn't (at least for a non-counter) one of those ways, even against a dealer's 6. Don't forget that it's a 20 and, by splitting, you have two hands of 10 which requires an additional bet. By standing, you expect to win 67.7% of all the money bet in that situation, but by splitting, your expected return drops to 55.6% or 27.8% per hand.

I am an average black jack player who has a few questions pertaining to double downs in basic strategy and card counting:

1) When you double down.. according to the chart you should always hit between 5 through 8...but why can't you double down on any value 11 and below when the dealer has a 5 or 6 facing up? The dealer will most likely bust anyway.. so wouldn't make sense to double down on your hand that's 11 and below?

Unfortunately, the dealer will bust with a 6 up only 42% of the time; the other 58% of the time, s/he will make a hand. By doubling on a 7 , for example, you'll lose 12.6% of all the $$$ you bet in that situation (assuming a 6-deck game, dealer stands on A-6), but if you hit, you'll win 3.4%, a difference of 16%. Believe me, the good people who figured out the proper playing strategy took all those situations into consideration.

2) When card counting. How do you know how many chips you should place as a bet. Is there a system or a formula that calculates how many units you should raise your bet (considering you are using 6-8 deck of cards). For instance when you have a card count of +7 how many chips should you place down 6-8 decks are used)?

The bet is based upon several things: the total amount of your 'bankroll', the edge the casino has 'off the top' and the number of decks remaining in the shoe. For a thorough discussion on how to figure this, see the lessons on my site. They're on the Blackjack Page under 'Money Management'.

Hot Tip: Proper basic strategy at Blackjack says to HIT your 11 against a dealer's upcard of Ace, if the dealer is required to stand on A-6 and to DOUBLE 11 against an Ace6 if the dealer must hit A-6.

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