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Ask the Wizard #231

There is a “Bad — It’s the New Good” promotion running in the high-limit room at the Barona casino. If the dealer in blackjack gets a 7-card 21, then all players actively wagering at the table will get $500. Also, if the dealer gets an 8-card 21, then all players at the table will split a progressive jackpot, which starts at $25,000, plus players at the other high-limit tables will each get $500. What is the probability of these events, and what is the value per hand to the player?

Larry from Las Vegas, NV

The following table shows the probability of a dealer 21-point hand according to the number of cards and number of decks.

Probability of Dealer 21-Point Hand

Cards 1 Deck 2 Decks 6 Decks
2 0.0482655 0.0477969 0.0474895
3 0.0537557 0.0530246 0.0525656
4 0.0184049 0.0184945 0.0185388
5 0.00310576 0.00326001 0.00335881
6 0.000291717 0.000344559 0.000380387
7 0.0000160093 0.0000234897 0.000029251
8 0.000000456411 0.000000997325 0.00000152356
9 0.00000000466991 0.0000000239012 0.0000000526866
10 0.0000000000064214 0.000000000262229 0.00000000115152
11 0 0.0000000000009179 0.0000000000148827
12 0 0 0.0000000000001003
13 0 0 0.0000000000000003

The next table shows the value in cents of the three prizes. The row for the 7-card prize is the value per hand of the $500 bonus for a dealer 7-card 21. The row for the 8-card prize is the value per hand of a $25,000 prize for a dealer 8-card 21. That should be multiplied by the ratio of the current jackpot to $25,000, for the value at any given moment. The row for the envy prize is the value per hand dealt at all other tables in the room of the $500 prizes for the jackpot hitting at another table.

Value of Prizes per Hand Dealt

Prize 1 Deck 2 Decks 6 Decks
7-card $500 win 0.80¢ 1.17¢ 1.46¢
8-card $25,000 win 1.14¢ 2.49¢ 3.81¢
8-card $500 envy bonus 0.02¢ 0.05¢ 0.08¢

Assuming a total of 8 active tables in the room, and 60 rounds per hour, and a $25,000 jackpot, the value of this promotion is $1.26 per hour at a single-deck table, $2.41 at double-deck, and $3.48 at six-deck.

How would I compare the actual hold percentage on each table game in comparison to the house advantage for the games to see if they are performing to their expectations? Is there a formula that I could use? For example, we know that the house edge on Roulette is 5.26% and that our hold percentage is at 25%. I am having trouble correlating the two, and coming up with a reasonable explanation for all games. It seems the more I read on it the more confusing it gets. Any help would be appreciated — there must be a formula to somehow mesh the two together.

Robert from Lawton

For the benefit of other readers, the house edge is the ratio of expected casino profit to the original wager, and the hold is the ratio of actual casino profit to chips purchased. The hold will usually be much higher, because over time the same chips will circulate back and forth. The longer the player plays, the more the house edge will grind down those chips, resulting in a greater hold, but an unchanged house edge.

There is no formula expressing a relationship between house edge and hold. To get from one to the other you would need to know how much the players bet, how well they play, and how long they play. I have said this many times, but I don’t understand why casino management cares so much about the hold percentage. What should matter at the end of the day is the hold, or the actual profit measured in dollars.

Why does your hand ranking chart differ from the one by David Sklansky (published in his book Hold ’Em Poker for Advanced Players, which also appears in the Wikipedia entry for Texas hold ’em starting hands? For example, Sklansky ranks 76 suited and A9 offsuit equally with a 5 rating. Your chart ranks 76 suited as an "11" but ranks A9 suited as a "16"! Care to explain why there are these discrepancies?

John from New York

I think you misread Skanskey’s table. He rates 7-6 suited equally with A-9 off-suited with a 5. I rate 7-6 suited with an 11, and A-9 off-suited as a 10. So we both put them about the same.

What percentage of rolls in craps are come out rolls?

Eddie from Seattle

The expected number of total rolls is 1671/196 = 8.5255. Interestingly, the expected number of rolls for a point is exactly 6. That leaves 2.5255 come out rolls. So the percentage of come out rolls is 2.5255/8.5255 = 29.6%.

I’m shopping around for a mortgage. One company is offering an interest rate of 5.75%, plus one point, on a 30-year fixed. Another is charging 5.875% without a point. Which is the better offer?

Dorothy from Miami, FL

For the benefit of other readers, a point is a commission charged for the loan. For example, on a $250,000 loan one point would be $2,500. I’m going to assume that the borrower would add the point to the principal balance, and never pay down the principle early.

The following table shows the equivalent interest rate without the point, according to the interest rate with one point and the term.

Equivalent Interest Rate with No Points

Interest Rate with One Point 10 years 15 years 20 years 30 years 40 years
4.00% 4.212% 4.147% 4.115% 4.083% 4.067%
4.25% 4.463% 4.398% 4.366% 4.334% 4.318%
4.50% 4.714% 4.649% 4.617% 4.585% 4.570%
4.75% 4.965% 4.900% 4.868% 4.836% 4.821%
5.00% 5.216% 5.151% 5.119% 5.088% 5.073%
5.25% 5.467% 5.402% 5.370% 5.339% 5.324%
5.50% 5.718% 5.654% 5.621% 5.590% 5.576%
5.75% 5.969% 5.905% 5.873% 5.842% 5.827%
6.00% 6.220% 6.156% 6.124% 6.093% 6.079%
6.25% 6.471% 6.407% 6.375% 6.344% 6.330%
6.50% 6.723% 6.658% 6.626% 6.596% 6.582%
6.75% 6.974% 6.909% 6.878% 6.847% 6.834%
7.00% 7.225% 7.160% 7.129% 7.099% 7.085%
7.25% 7.476% 7.412% 7.380% 7.350% 7.337%
7.50% 7.727% 7.663% 7.631% 7.602% 7.589%
7.75% 7.978% 7.914% 7.883% 7.853% 7.841%
8.00% 8.229% 8.165% 8.134% 8.105% 8.093%
8.25% 8.480% 8.416% 8.385% 8.357% 8.344%
8.50% 8.731% 8.668% 8.637% 8.608% 8.596%
8.75% 8.982% 8.919% 8.888% 8.860% 8.848%
9.00% 9.233% 9.170% 9.140% 9.112% 9.100%
9.25% 9.485% 9.421% 9.391% 9.363% 9.352%
9.50% 9.736% 9.673% 9.642% 9.615% 9.604%
9.75% 9.987% 9.924% 9.894% 9.867% 9.856%
10.00% 10.238% 10.175% 10.145% 10.119% 10.108%

This shows that a 5.75% interest rate with one point is equivalent to a 5.842% with no points. In other words the payment would be the same both ways, assuming the point charged is added to the principal balance. Your other offer was 5.875% with no points, which is higher than 5.842%, so I would take the 5.75% with the point.

P.S. For those of you wondering how I solved for i, I used the rate function in Excel.