I have a good friend who is well aware that the house has an edge at blackjack but looks for the lowest stakes table so he can play for quite a while before he is at substantial risk of running out of money. He considers it good value entertainment. That makes perfect sense to me, but I wanted to estimate what he is paying for his entertainment. Here is my estimate.
Optimal Strategy
When I watched my friend play at a casino, they were shuffling six decks together. This would make card counting essentially infeasible. Given that, previously played cards or others’ players cards add no useful information. The likelihood of the next card turned is (close enough) the natural probability: each card has a 1/52 chance.
For a basic game where the dealer has to follow a fixed rule in hitting his cards, but you have a choice, it wouldn’t be hard figure the optimal hit choice based on your current count and the card the dealer is showing (you want to maximize the probability you beat the dealer). Figuring that anything that straightforward would already have been done, I did some trivial web searching and ran across this suite of articles; they consider not only a simple game of blackjack but strategy for the more sophisticated options too (like splitting and doubling). I haven’t validated them, but they seem thoughtful and plausible. The house still has an edge, but at 0.5% it is quite small. (Slot machines have a typical house edge of between 5 and 10%.)
If you play any strategy other than the optimal strategy, the house edge will increase, and if you play your hunches (play the optimal strategy but occasionally deviate based on emotion), the house edge will increase. Let’s assume you stick with the optimal strategy.
Side Bets
It seems that casinos use some of their real estate on the table for side bets like a standalone bet that the dealer goes bust. Since casinos are run by smart and experienced people who want to make money, we can assume that the casino would not make them available unless they had at least as high a house edge as the basic game; otherwise it would not allow the distraction and loss of time from explaining, placing, and paying out the side bets and would instead keep all activity focused on maximizing the frequency of the main game. Moreover, since the casino would have a choice of side bets, it will use table real estate for the ones for which the product of house edge and player propensity to make the bet are maximized (that is, have something that is significantly in the house’s favor and likely to attract a “hunch” play). They will have had the opportunity to experiment with many side bets over time and will have settled on the ones that are most profitable per unit time to the house. I think we could prove this by examining each side bet and the payouts they offer, but I’m going to take it as convincing enough in principle to not bother doing that.
Let’s assume you don’t make any side bets.
Cost of Entertainment
Let’s assume you are making a fixed bet of B on every hand, and that each hand takes a time T to complete. The expected value (probability weighted average of all possible outcomes) for each hand using the optimal strategy with no side bets is that you will lose the house edge times your stake, or B/200, so your cost of entertainment per unit time is B/200T. So, for example, your bet is $2 per hand and a hand takes one minute, you are spending 0.01/minute or 60 cents per hour. That is pretty darn cheap.
Most people have a higher, sometimes much higher, cost of entertainment by doing one of several things:
- Increasing the bet above the minimum. Larger numbers at stake make it more exciting, but the cost of entertainment goes up proportionally with the stake, so it costs more too.
- Not playing the optimal strategy, varying from the optimal strategy occasionally based on hunches, or also playing side bets.
- Running out of capital – your capital will vary randomly – though the expected value decreases with each bet, the actual value could be as high as (roughly) your initial capital C + tB/T, where t is elapsed time, or as low as C – tB/T. Assuming you aren’t willing to buy more chips, you will stop if you exhaust your capital (this is called a stopping time). Since there is some finite probability this will happen and prevent you from relying on the laws of probability (law of large numbers) from getting you arbitrarily close to the expected loss per bet, it means that having finite capital increases your cost of entertainment.
Figuring Particular Winnings
I use a simplified model to estimate the odds of particular winnings after a number of hands. Extending the above example, I assume you bet B on every hand and you either win B or lose B (one or the other), with the probability set by the house’s edge.
Under this model, winnings are governed by the binomial distribution (the distribution of the number of heads in a total number of tosses of a possibly unfair coin). For a large number of tosses, we can approximate the binomial distribution with a normal distribution, but for a smaller number we can just calculate it directly. I used the BINOMIAL.DIST function in Excel.
Our winnings after n hands will be W = BX – B(n – X), where B is the size of the bet and X is a random variable with a binomial distribution of n trials and with p = 0.495 (the probability of winning a hand given the house’s edge). Alternatively, W = B(2X – n). In the following examples, I’m looking at 90 hands played (about an hour and a half at one hand per minute).
Example 1 – What are the probabilities of being down, up, and breakeven after 90 hands?
I use Excel to figure the cumulative probability of 0-44 wins, and of exactly 45 wins. The probability of 46 or more wins will be one minus the sum of the other two. The result is:
| Probability of Being Down | 49.6% |
| Probability of Breaking Even | 8.4% |
| Probability of Being Up | 42.0% |
Example 2 – If the bet is $2 per hand, what is the probability of being up at least $24 after 90 hands? (I use 24 to make the target an integer number of $2 bet outcomes.)
We need to win 51 or more of our 90 hands to be up $24 or more. Using Excel, the probability of 50 or fewer wins is 0.895, so the probability of 51 or more wins is 0.105, or about 1 in 10.
Example 3 – What is the probability you will have lost an initial capital of, say, $100 after 90 hands at a bet of $2 per hand?
For simplicity, I assume that if you went bust before 90 hands, you borrowed money to keep going. Under this assumption, you will need to have lost at least 50 more times than you won in your 90 hands (in any order) to be $100 down or more. In other words, you will have to have lost 70 or more times out of your 90 hands.
Using Excel, the probability of this is 0.1% (about 1 in 1000).
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