Theoretical Probability Of Winning Craps

What are the odds of winning at craps? At the meeting we worked out the following reasoning: There is a 2/9 chance of rolling a natural and winning in the come out phase. There is a 2/3 chance of rolling a point. The chances of making the point are written above. Stephen had the idea of using the average chance of making the point – 4/10. Gives the best odds of winning among all casino games. As a matter of fact, as we will show in this article, among all the different bets one can place in the game of craps, the Pass Line bet regards one of the best bets due to its high winning probability. In this article, we will concentrate on deriving results for a Pass Line bet.

Casino gambling can be fun, exciting, and a little nerve-wracking. Knowing which casino games have good odds in your favor (gamble on these!) and which ones to avoid will help you to keep your cool — and as much of your money as possible. By making your money last, you extend your playtime; so manage your money wisely.

Casino Gambling Games with Decent Odds

For gamblers, four casino games offer decent odds. That is, if you made a hundred $1 bets at these recommended casino games, your average loss would only be about a buck. These odds aren’t good enough to retire on, but they stretch your bankroll as far as possible, even if you’re on a limited budget.

Neither life nor gambling offers any guarantees, and your actual daily results may be far different than your theoretical loss, so plan accordingly.

Jacks or Better Video Poker (the 9/6 Version)

The 9/6 version of Jacks or Better is considered full pay because you get a payout of 9 for 1 for a full house and 6 for 1 for a flush. The 9/6 version is the only one whose payout rate makes it worthwhile to play. Tips for playing this game well include:

Make sure you’re playing a full-pay 9/6 version by looking at the Full House/Flush payouts for one coin.
Because of the increased payouts for a royal flush, always play the maximum coins to get the best odds — even if that means finding nickel or quarter machines to stay within your bankroll.
Never keep a kicker (a high card along with a pair).
Throw away a small pair (tens or lower) if you have four cards to a flush or three cards to a royal flush.

Blackjack

The way to win at Blackjack is to get better cards than the dealer. The aim is to have your cards add up to a maximum of 21.

Play the table with the least number of decks.
Stay away from gimmick games like 6-5 blackjack, Spanish 21, or Super Fun blackjack.
Remember the following important tips
Double down on 10 or 11 when dealer shows 2 to 9. (To double down means you double your bet and are allowed just one more card.)
Stand on 13 to 16 when dealer shows 2 to 6.
Hit on 12 to 16 when dealer shows 7 or higher.
Never take insurance.

Craps

Rolling two dice is the essence of craps and what number those dice add up to determine whether bettors on the Pass line win or bettors on the Don’t Pass line do. A new player’s first roll is her come-out roll, and other players wager on what the shooter will roll. The smartest strategy is to stick to the following bets in craps:

Pass line bet with odds bet
Come bet with odds
Don’t pass bet with odds
Don’t come bet with odds

Baccarat

You may be intimidated by baccarat with its glitz, glamour, and European aura, but behind the stiff exterior lies a decent game for the average bettor because there are only three possible results of each hand and there’s no skill involved — it’s strictly the fall of the cards. A couple tips for doing well:

Always bet on the banker (house edge is 1.06 percent, meaning if you made a hundred bets of $1, your average loss would be about a buck).
You can also bet on the player’s hand. (The house edge is only slightly worse than the banker’s hand at 1.24 percent.)

Bets to Avoid in Otherwise Safe Casino Gambling Games

Some casino games are safer to gamble on than others, but even these have versions or bets that’ll drain wallets. Avoid these casino game versions or bets at all costs:

Jacks or Better Video Poker (6/5 Version): The payouts are much worse on some versions of video poker. For example, when you make a full house on a 6/5 machine, you get paid only six coins instead of nine. Play only the 9/6 version of Jacks or Better.
Blackjack — the insurance bet: This side bet pays back 2 to 1 when the dealer has a blackjack, but the true odds are closer to 2.2 to 1, making this a losing bet over time. You’re gambling, after all, you’re all about the risk, so forget the insurance.
Craps — the field and proposition bets: The house edge can be as high as 16 percent for these seductive bad bets. (Can you say ouch?) Stick with Pass, Don’t Pass, and Come bets.
Baccarat — tie bets: If you want to keep your money, steer clear of this bet; the house has an edge of 14.36 percent on tie bets. You want to enrich yourself, not the house.

Casino Gambling Games to Avoid at All Costs

If you’re gambling in a casino, you know that the odds always favor the house. But some casino games have higher odds for the house than others, including the ones in the following list. Don’t gamble on these games unless you want to go home with a lighter wallet:

Three-Card Poker
Let it Ride
Red Dog
Roulette
Caribbean Stud Poker
Big Six Wheel
Keno

How to Manage Your Money at a Gambling Casino

Successful casino gamblers understand the math and odds behind the games, but discipline with money is also critical. Managing your money at a casino requires self-control. Use these tips to keep yourself in the safe zone:

Never borrow money while gambling. Chances are good that you’ll lose it, making a bad situation even worse.
Only bet what you can afford to lose. Gambling with money you’re scared to lose is neither enjoyable nor wise.
Set and stick to a budget. Write it on your forehead if you have to, but no matter what, when you hit that number, leave the casino.
Avoid ATM machines or lines of credit. Using credit is a really easy way to get financially overextended while gambling.
Limit your playing time. Taking a break keeps your mind sharp.
Hotel rooms have beds — use them. When you only gamble a few times a year, it’s tempting to play marathon sessions. But your body and your bankroll aren’t built to handle the casino’s pulsating 24/7 schedule, so make sure you sleep and rest.
Don’t drink and gamble. If the casino serves drinks for free, you should be very suspicious.
Remember the law of gravity — what goes up must come down. Casino profits soar when players get greedy. Quitting with a small win is far better than losing your shirt.
Don’t play any game you don’t understand. One of the fastest ways to burn through your cash is to jump into a game where the rules or strategies confuse you.
Never lie about your wins or losses. If you find yourself deluding a person or several people in your life about your gambling, take pause and reevaluate your gambling.

Act One: Launch

We began by watching a clip from the move, A Bronx Tale. (Be warned: there is some… colorful language in this clip)

The Kid’s First Roll

After watching the clip two times, we had the following questions:

Why do they have the kid rolling? Is it luck?
What is the mathematical value/concept of being a jinx?
How does the betting work in craps?
What is the difference between craps and Cee-lo (another dice game popular with Leo’s students)?
What is he going to roll?

All interesting questions, but for today we focused on the last one.

Theoretical probability of winning craps games

Act Two

Mark then asked, “What do you need to know from me in order to answer that question?”

What are the rules of Craps? – Basic rules of craps

What phase of the craps game is this – the come out phase or the point phase?

This clip is of the come out phase

Act Three and Extension

Here are the possible results during the Come Out Phase:

There was a fire alarm in the middle of the meeting and we had to evacuate and continue the meeting in the courtyard. As a result, we didn’t get to watch the second video, which picks up the action right where the first video leaves off.

Will He Make the Point?

The first video focuses our attention on the Come Out phase, but the second video focuses on the Point phase. Even though we didn’t get to see the second video, people’s curiosity about the Point phase led to some interesting discoveries.

One challenge for figuring out the probability of winning the point phase was figuring out the number of possible outcomes. After some debate, we decided it is no longer out of 36 outcomes as it had been in the Come Out phase. The only rolls of consequence are rolling a 7 or making the point. Any other roll results in just rolling again. That means the only outcomes to consider are the six 7s and however many ways there are to make the sum of the point the shooter rolls in the Come Out Phase. For example, if you roll a 4 in the come out phase, there are 9 possible outcomes of consequence for you in the point phase – the six 7s and the three ways to make a 4.

What are the odds of winning at craps?

At the meeting we worked out the following reasoning:

There is a 2/9 chance of rolling a natural and winning in the come out phase.
There is a 2/3 chance of rolling a point. The chances of making the point are written above.
Stephen had the idea of using the average chance of making the point – 4/10
If there is a 2/3 chance of rolling a point and then a 4/10 chance of making the point (and winning), then there is a 4/15 (or about 27%) chance of winning by rolling and then making a point. Add that to the 2/9 chance of rolling a natural and you have about a 48.8% chance of winning at craps.

A second reasoning:

After the meeting, Mark and Eric were talking about whether or not we need to somehow weight the different probabilities of rolling a point in the Come Out Phase. In the reasoning above, we use the overall chance of rolling a point (2/3 or 67%). The work below is still an estimate, but tries to incorporate the idea of weighting the different possible point rolls differently. Based on the calculations, there is an ever better chance at winning at craps – a little over 49%

Additional Resources

Possible Sums in Craps: This is the blank handout Mark gave out at the meeting. It is one way to organize and see all of the possible sums.

Roller Derby: This is a great activity for having students analyze a game of chance involving rolling two dice.

Sums of Two Dice – an interesting photograph of dice to inspire some mathematical discussion

Two Videos on How to Play Cee-lo (a game Leo’s students enjoy)

Gamblers, Scientists and the Mysterious Hot Hand A recent article in the NY Times about a lot of things, including research into the human mind’s response to randomness, especially as manifested as a gambler’s fallacy or a hot hand fallacy.

It Was a Good Day, by Ice Cube (references a winning game of craps)

Dollar Bill Blues, by Townes Van Zandt (“I’ve always been a gamblin’ man, roll those bones with either hand, 7 is the promised land…”)