Ask The Wizard #31

Has anyone done work on the optimal betting strategies in "Let It Ride" w/ additional information? I know the casinos say players may not look at other players hands, but in practice, it seems that most players know what at least two other players are holding. I may want to go for an inside straight if I know that at least six of the cards on the table are NOT what I need to pull.

Brian from Stafford, Virginia

Yes! In Mastering the Game of Let It Ride by Stanley Ko a section is devoted to this topic. Ko explains how the odds change if you have a 4-card straight or flush and can see extra cards. He does not indicate such adjustments at the three-card stage of the game. You can find this booklet at that Gambler's Book Club.

Hey wiz, Great Site. I wondered if you knew the House Odds on a game played at Foxwoods in Connecticut, called "Catch A Wave."

Ken L. from Boston, USA

Ask and ye shall receive. Please see my page on Catch a Wave.

Is my boyfriend cheating on me?

Melevinna from IA City, USA

How should I know? However don't make any accusations unless you have sufficient evidence to back them up.

My question revolves around the system Norman Leigh used in the 60s to break the bank in Nice. His team used a reverse Labouchere system, which involves absorbing a number of small losses before hitting a big win (based on the idea that instead of canceling out wins they are added to the sequence and losses are crossed out ensuring that each game can only lose a fixed amount but can potentially win the table limit). Other than Norman's book (and books based on it) I have not seen any analysis of the system. The book states that this approach uses the house edge against them and that in the long run the player will come out ahead. Is this nonsense or is there something in this idea?


If, indeed, they won it was because of luck and not because it was a winning system. As I have said a thousand times before, any system based on a negative expectation game in the long run not only can't overcome the house edge, it can't even dent it.

First, let me congratulate you on your great website. I tell everyone I know that if they are going to gamble to make sure that they visit your site first! My question is on Three Card Poker ante & play. If you know one of the dealer's three cards how should you change your basic strategy and could you obtain an advantage over the house and by how much?

G.M. from Monmouth County, New Jersey

Please see my hole carding strategy for Three Card Poker. Following the strategy, you will enjoy a 3.48% advantage!

Concerning the 5% vig on buy bets and lays how would the odds change if $1 was charged for $20-$39, $2 for $40-$59, $3 for $60-$79 and $4 for $80-$99 without round up. Your information you give is outstanding.

Bry from Chesterton, USA

Thanks for the compliment. The formula for the house edge in buy and lay bets is the commission divided by the bet plus commission. In this case, the best bet is to bet $39 for the $1 commission. On the buy bet the house edge would be 1/40 = 2.5%. Assuming you can lay $78 to win $39 on the 4 and 10, and still only pay $1, the house edge would be 1/79=1.27%. I'll leave the other situations as an exercise for the reader (I hated it when my math books would say that).

How do you win money playing solitaire in Vegas?

Pattie from Arlington, USA

I have never seen solitaire played for money in Vegas. I understand in the early days of Vegas people wagered on the standard Klondike variation of solitaire but I don't anything else about it.

In general, when betting on anything which pays even odds, is there any kind of "system" to help improve chances and or payoffs?

JJ from Boston, USA


What is the probability of getting a three pair in Pai Gow Poker? Are the chances lesser or greater than three of a kind?

Alex from Long Beach, Mississippi

Not counting a three of a kind and two pairs, the following are the ways to get a three pair and number of combinations.

No wild card: combin(13,3)*10*63*4 =2471040
Wild card used to compete pair of aces: combin(12,2)*10*62*42 = 380,160
Wild card used as singleton ace: combin(12,3)*63 = 47,520

The total number of combinations is 2,898,720. This is less than half of the 747,0676 combinations for a three of a kind.

I've noticed some new video slot machines (Money to burn, High Bid, Money for nothing, Who Dun it, etc) that differ from the normal three-reel slots in the following ways -- first they have five reels. You can typically bet on 1 to 9 pay-lines (even though some have as many as 15 different pay lines), and multiple coins per line; thus, with nine pay-lines and five coins played per line, you would have a total bet of 45 coins (even in nickles, this can start to add up!). Most payoffs are multiples of the line bet, even though there are some "bonus" wins that pay multiples of the total amount bet. Is it best to always pay all possible pay lines, or is there an optimum combination of pay lines to play to achieve the best return? I suspect that getting a winning combination on any particular pay line is the same for all, but wondered if you have any better insight to share.

Scott from Leawood, USA

Each frame in these video slots is weighted equally. Any given line is equally likely to produce any given combination. Thus, the return is the same regardless of the number of coins played.

I wonder if you could comment on the Casino practice of switching dealers. It always seems to happen that the table is on a winning streak and then the casino switches dealers mid-shoe. All of a sudden, everyone starts to lose. Do you think that certain dealers tip the balance more in the casino's favor?

Hobbes from Toronto, Canada

The casinos switch dealers when it is time for someone to go on a break or go home. Switching dealers does not change the player's odds unless the player is a card counter and the game is single- or double-deck, where a new dealer necessitates a fresh shuffle.

How many numbers does the RNG (Random Number Generator) pick for each spin in a slot machine? Is it three numbers (1 for each reel) or is it 1 number that's mapped to a unique combination of symbols for all 3 reels?

Vincent from Chicago, Illinois

The machine picks one number for each reel.

Great site, Mike! Often times I hear the word, "binomial distribution" being used in gambling. Can you explain to me what it means? Thanks in advance.

Dennis from Toronto, Ontario

Thanks for the compliment. Any introductory probability and statistics book should give good treatment to the binomial distribution. Briefly, the binomial distribution is the probability that any given number of events will happen given a specific probability for each event and a specific number of trials. Specifically if the probability of each success is p, the number of success is s, and the number of trials is n then the probability of s successes is ps * (1-p)n-s * combin(n,s). The combin function is explained in my glossary. For example, suppose you want to know the probability that in 100 spins of a roulette wheel the number of reds will be exactly 60. According to the binomial distribution, the probability is (18/38)60 * (20/38)40 * combin(100,60) = 0.003291.

Excel also has a function for the binomial distribution. It is =BINOMDIST(x,n,p,0), where:

x=number of positive trials. n=total number of trials. p=probability of success in any given trial.

Use a 0 in the fourth position of the function for the exactly probability of x wins. For the probability of x or less wins, use a 1.

In the roulette example above, the function would be =BINOMDIST(60,100,18/38,0)

I just finished reading your section on strategy for craps with great interest. I understand that better the pass line and come bets with full odds is a good strategy. My question is "does the house edge change at all when playing a strategy of pass line with full odds and making a maximum of two come bets with maximum odds?" In other words, how does time (more rolls) and having more money at risk affect these odds if at all? Or should a person stick with just the single bet with full odds? This seems to be a favorite strategy for most knowledgeable players I have met at the dice tables.

Jim from Yakima, USA

The house edge is the same regardless of how many come bets you make assuming you always take the maximum allowable odds and leave the odds turned on during a come out roll. How many come bets you make should be a matter of personal preference.