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Ask the Wizard! (No. 134)

April 3, 2005 Column

Last night a player offered me a side wager in Texas Hold 'em. He said that at least one face card (or any three ranks) would appear on the flop and offered even money? Should I have accepted the bet?

The number of ways you can choose 3 cards out of the 40 non-face cards is (40*39*38)/(1*2*3) = 9880. The number of ways you can choose 3 cards out of 52 is (52*51*50)/(1*2*3) = 22100. So the probability of not getting a face card is 9880/22100 = 44.71%. Thus the probability of getting a face is 55.29%. His side of the bet had a 10.58% advantage.

At the Borgata casino in AC, they have a different rule regarding commission for pai gow poker than the rest of the casinos. They only charge commissions in 50 cent increments, rather than the 25 cent norm. When I played and won a $25 hand, I only paid commission of $1. To what degree does this lower the house edge?

From my pai gow poker section we have the following probabilities.

Player wins both 28.61%
Tie 41.48%
Banker wins both 29.91%

So you are getting away with a 4% commission. As player your expected value is .2861*0.96 - .2991 = -2.44%. As banker (going one on one) is .2991*0.96 - .2861 = 0.001036. So a 4% commission lowers the house edge by about 0.3%.

P.S. In May, 2008, a reader wrote to say that the Borgata does use quarters in pai gow poker, and forces bets to be in increments of $5.

By what method do casinos pay you when you cash out? For example, if you were to win say $10,000-$15,000 playing Roulette or Black Jack can you get that money in a cashiers check, money order, etc.? As one certainly doesn't want to be walking around with or driving back to Canada with a bunch of cash on them!

I believe the policy at most casinos is that for large transactions you can have the funds any way you want. Before you consider laundering money by turning cash into checks be aware that casinos ask for a Social Security number and make a record of any transaction involving $10,000 or more.

A colleague of mine was recently playing at [an online] casino playing 10-line Jacks or Better Video Poker. Money was deposited and 10 hands were played. All 10 hands (and thus all 100 lines) failed to bring up a single win. Please can you calculate the probability of drawing a blank on 10 hands of 10 line JoB. Also, would the probabilty you calculate be evidence of a rigged game? Thanks in advance and keep up the (very) good work.

Here is the probability of winning zero per game according to the number of plays.

Probability of Winning Zero in n-play Video Poker
Plays	Probability
3	0.26260274
5	0.1301204
10	0.02591377
15	0.00649444
25	0.0007854
50	0.00002178
75	0.00000076
100	0

The table is based on a random simulation. I know it is theoretically possible to get a win of zero in 100-play, but in 15,820,000 games it just never happened. So please don't write about that. The table shows the probability of getting zero in 10-play is 0.025914, or 2.59%. The probability of this happening ten times in a row is 0.025914¹⁰ = 1 in 7,323,073,295,177,980.

I tried the software in question in free-play mode and my results seemed fine. In particular in 10 games I won something every time. However as far as I know no casino offers this software and takes real money players from the U.S. I'll plan to do some further investigating but don't want to explain how in this forum.

What is the added house edge in blackjack if players are never allowed to double nor split?

No splitting costs the player 0.58%.
No doubling costs the player 1.47%.
No splitting nor doubling costs the player 1.91%.

What is the probability of rolling 13 or more with 3, 4, and 5 dice, if you are allowed to keep the highest three dice in your roll?

Here are the probabilities:

3 dice: 25.93%
4 dice: 48.77%
5 dice: 66.13%.

I've seen video blackjack in several LV casinos and am wondering if the game is regulated in a similar fashion as video poker. If so, would playing the games using basic table blackjack strategy give the player the best payback percentage? If not, is there a basic strategy for video blackjack?

Yes. Any video representation of a card game in the state of Nevada must be dealt from a fairly shuffled deck. In other words you should expect the same kind of outcome as in a live game with the same rules. So basic strategy tables will also work for video blackjack.

[Bluejay adds: Every video blackjack game I've seen pays only even money on naturals, which significantly increases the house edge.]

Last night I was at a casino in Louisiana and after playing for a while I noticed that the Big 6/8 bet was replaced with an Over/Under 7 for one roll bet. The crew member said it was new and paid even money. He then mentioned that he thought the field was a better bet. Was he right?

No! The probability of winning is 15/36 = 41.67%, for a house edge of 16.67%. Even if the 2 and 12 both pay 2 to 1 on the field the house edge is only 5.56%. The over/under 7 bet ties for the any 7 bet as the worst bets on the table.

What is the probability of getting the "dead man's hand", a two pair of aces and eights?

There are six ways to arrange two suits out of four for each pair. Then there are 44 cards for the singleton. So the number of successful combinations is 6*6*44 = 1584. There are 2,598,960 combinations in total, so the probability is 0.0609%.

If I make a maximum bet, can I still make a bet for the dealer?

Here in Vegas, yes you can.

I was curious. How do the odds change in video poker if a person always shoots for a natural royal flush? (In other words always holding the most beneficial hadn to obtain a natural royal flush.... disregarding all other possible hands.)

A strategy of going for a royal at all costs, as if all the other hands paid zero, would result in a return of 47.85% on a 9/6 Jacks or Better game. The expected frequency of a royal would increase form once every 40388 hands to once every 23081.

Closing Words

This will be the final "Ask the Wizard" column for the indefinite future, and probably ever. Between February 5, 2000 and April 3, 2005 I have answered your questions, meanwhile writing 134 columns. The vast majority of questions I answered never went into the column. Unfortunately the volume of questions became too unbearable. Despite my urging that questions be brief and cover matters not already addressed on the site I received dozens of e-mails daily requesting math tutoring or personal gambling advice. If I tried to give a brief reply out of politeness I was often insulted in return because my brevity was insufficient. Answering questions became too much of a chore and I feel my energies are better spent refining existing content on the site and adding new material.

It is with much regret that I have to end the column after five years. I do appreciate all the compliments and kind words I have received over the years. While I am accepting no questions I still accept comments and corrections. Thank you for your understanding.

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