Ask the Wizard #173

Thanks for the compliment. There is a formula for questions like this, which I explain on my site www.mathproblems.info, see problem 116.

With a bankroll of b units, winning goal of g units, probability of winning p, and probability of losing q your probability of success is ((q/p)^b-1)/((q/p)^b+g-1). In this case b=20, p=18/37, q=19/37, and g=20, 50, and 100. So for a bankroll of $200 the probability is ((19/18)²⁰-1)/((19/18)⁴⁰-1) = 0.253252.

For a winning goal of $500 the probability is ((19/18)²⁰-1)/((19/18)⁷⁰-1) = 0.045293.

For a winning goal of $1,000 the probability is ((19/18)²⁰-1)/((19/18)¹²⁰-1) = 0.002969.

Thanks. To answer your question, no, I don’t think it is a good game. The house edge is too high. If you are looking for a big win you could play a progressive betting system in blackjack, pressing your bets as you win. Of course this comes at the cost of frequent smaller losses.

Thanks. Payout percentages such as this are historical. For example King Neptune's casino posts their June 2006 report on their web site. The 96.78% for table games means that in June 2006 the ratio of money returned to money bet was 96.78%. In other words an actual house edge of 3.22%. Your own results will depend on the game rules, your skill (in games of decision making), and luck. In most games the odds are quantifiable so payout reports are not useful. It shouldn't matter to you how badly other players have played or the mix of games they chose. Where these reports are very useful is in evaluating the slots. No casino that I know of volunteers how loose their slots are theoretically set, but such payout reports gives the user a good idea. If looking at other months you see that King Neptune's pays about 96% in slots. I also think it is a good sign of a good operation to have return percentages independently verified. It shows the casino has nothing to hide.

Most redoubling situations tell you to double anyway. However, with a soft 15 to 17 against a 3, when the strategy says hit, you should actually redouble.

I asked my father this question since he has a Ph.D. in physics and also a solar panel on his house. Here is what he said,

It would help, but the economics might not justify it. Probably less than 25% of heat enters houses through the roof. The reflectivity of the mirrors would probably degrade to 60% or less as they age and get dirty. It makes a lot more sense to use that space for water heaters or solar electric panels. On a sunny day, my roof panels provide enough power to run both the A/C and the pool pump, which are my biggest power eaters. When one or both are off, my meter runs backwards. The pool heater panels had the pool temp up to 90 degrees last week. I had to cut back on the pumping time.

Thanks. Paying a 25-cent commission on a $3 bet amounts to an 8.33% commission. Assuming you only play as the player, you will win both bets 28.61% of the time. So the normal cost of the 5% commission rule is 0.2861×0.05=1.43%. The losing on copies rule costs the player 1.30%, for a total house edge of 1.43%+1.30% = 2.73% normally. In the case of this game, the cost of the commission is 0.2861×0.0833=2.38%. So the total house edge is 2.38%+1.30%=3.68%.

As my blackjack section shows, the 2 to 1 on blackjacks is worth 2.27% and doubling on 3 cards is worth 0.23%. Otherwise the rules look standard. All things considered, the house edge in the blackjack game has a player advantage of 2.1%. The probability of winning on a 4 or 10 in craps is (6/36)×(3/9) = 5.56%. Every time this happens you get an extra unit, so it is worth 5.56%. Normally the house edge on the come bet is 1.41%, so overall the player edge under this rule is 4.15%. So I agree that craps was the better game to play.

A player may call for less in that situation, but he may only win the amount of his call bet from each of the other players who posted the small blind. In the example above, 15 chips from each player posting the small blind would be put in the pot. All other money would be put in a side pot, which the short-stacked player would not be eligible to win.

My thanks to Ashley Adams for her help with this question.

I forwarded your question to Brian, a former regulator and current casino manager. Here is what he said.

There are two types of changes. The first would involve completely swapping out the machine and the second would consist of simply changing the game, but keeping the existing cabinet. As you can imagine, changing the software is much cheaper which is why there is so much hype around downloadable games. How often games are swapped out depends on a casinos capital expenditures budget. Participation machines are turned over much more rapidly because the manufacturer has a vested interest in keeping the best product on the floor. In many instances, they will handle the scheduling for software and new machine replacements. Participation machines are those that are on lease to the property by the manufacturer. Usually, the manufacturer gets 20% of the revenue, less taxes. From an accounting perspective, the useful life of a slot machine is 5 years and then the asset is fully depreciated (no longer has a book value). The final consideration is popularity. How often do you go into a casino and see a section of slot machines that are the old IGT three reel Red White and Blue machines? If the machines are performing well, why spend $10,000+ to replace each unit?

Assuming six decks, the probability of a suited blackjack is 4×6×24/combin(312,2) = 1.19%. So the $25 bonus on that is worth $25×0.0119 = $0.2968 per hand. The probability of a suited 7-7-7 is 4 × combin(6,3)/combin(312,3) = 0.000015957. So the value of $100 on that is $0.0016. The probability of suited 6-7-8 is 4×6³/combin(312,3) = 0.00017234. So $100 on that is worth $0.0172. The probability of a suited ace and jack of spades is 6×6/combin(312,2) = 0.0007420. So $100 extra on that is worth $0.0928 (the player is already getting $25 for the suited blackjack). Adding this all up, the bonuses are worth 11.25 cents. So this is nowhere near enough to compensate for the 50-cent commission.