Length of a Craps Roll - Again!

I recently received the following email message:

Dear Don,

A while ago I read that the average number of rolls until a seven-out in craps is around 8.5. I was a math major in college and decided to try and see if I could verify this myself. I set up an eight state Markov chain with one absorbing state, the seven-out. After doing the computations I got an answer of about 6 rolls. My calculations seem correct to me. What is the correct answer? If it helps, I will send you my transition matrix so you can see what I did.

Thanks,
Josh

Well, Josh, there's good news and bad news. The good news is that I'm sure your calculations are all correct; you don't have to send me your transition matrix. In fact, I am impressed with the fact that you set everything up correctly. The bad news is that you solved the wrong problem, albeit, correctly. Let me explain.

My guess is that you calculated the cumulative probability of a seven out for 2, 3, 4,... rolls and discovered that after 6 rolls the probability of a seven-out was slightly better than 6/5 (Was your figure around 0.502789?). This means that the median number of rolls is just a hair under 6. If we don't split hairs, this means that half the time a seven-out will occur before 6 rolls and half the time a seven-out will occur after 6 rolls. You calculated the median rather than the mean.

In many applications the mean and the median are the same, so this is an easy mistake to make. It is, however, easy to see that they can be different. Consider the numbers 1, 2, 3, 4, 10 with each being equally likely. The median is 3 since half the other numbers are below 3 and half of the other numbers are above 3. On the other hand the mean, or average, of these numbers is 4.

So, Josh, give it another try; the 8.5 figure is correct. Here is a hint. Calculate the expected number of rolls for a Pass Line decision and the number of Pass Line decisions per seven-out. If you need some help, go into the archives on this site and look for my January and February 2000 articles; the February article should already look familiar to you.

If any of the rest of you readers have questions I can be reached at 711cat@comcast.net. See you next month.