**AND**2+1 as

*two different options*or if we should just count them

*together as one option*(which, in all honestly, I tried to intentionally bring out by giving students pairs of dice that were different colors).

The group that said we should count them as two different options ended up with 21 total possible outcomes and the following theoretical probabilities:

**2**

**3**

**4**

**5**

**6**

**7**

**8**

**9**

**10**

**11**

**12**

4.8% 4.8% 9.5% 9.5% 14.3% 14.3% 14.3% 9.5% 9.5% 4.3% 4.3%

The group that said we should count them together as one option ended up with 36 total possible outcomes and the following theoretical probabilities:

**2**

**3**

**4**

**5**

**6**

**7**

**8**

**9**

**10**

**11**

**12**

2.8% 5.6% 8.3% 11.1% 13.9% 16.7% 13.9% 11.1% 8.3% 5.6% 2.8%

Neither side was willing to budge, so I suggested we conduct a HUGE experiment with LOTS of trials across all three of my classes so that we could put the results together and see what conclusions we could draw. We did something like 3,500 trials...and here is what we found (experimental probabilities in brown on the far right):

I was reminded of a quote from Les Steffe's work that I read a while back:

"A particular modification of a mathematical concept cannot be caused by a teacher any more than nutriments can cause plants to grow. Nutriments are used by the plants for growth but they do not cause plant growth."