Three-Act Math: Probability in Craps

What are the odds of winning at craps? Is craps a fair game? What’s your chance of making the point? A three-act math task inspires some questions in probability.

Facilitator(s): Mark
Date of Meeting: October 13, 2015
Problem: · url

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:

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

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

Act Two

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

  • 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
  • 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.
  • Zaption is the website Mark used to edit the videos from the movie.

In attendance: Cynthia, Eric, Esther, Leo, Linda, Maritza, Mark, Michael, Solange, Stephen

Programs represented: The Literacy Assistance Center, Turning Point, CUNY Central, LaGuardia Community College, Borough of Manhattan Community College, Touro, Y-Roads Center Bronx

Location: The Literacy Assistance Center at 445 5th Avenue

Respectfully submitted by Mark

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