What's the Chance?

We've had a lot of fun this week exploring probability during math workshop. We started the week throwing wadded-up pieces of paper in the trashcan (you don't get to do that too often in school, do you?). We found you are certain to hit the trashcan from 3 cm away, likely to hit the trashcan from 150 cm away, unlikely to hit the trashcan from 250 or 350 cm away, and it's nearly impossible to hit the trashcan from 6m away with your eyes closed.

Our second activity this week dealt with dice. We posed the question: If you roll two dice and add them together, what sum will you most likely roll? There were a variety of predictions to this exploration, and most of us were surprised by what we found. Partners worked together to roll dice and record their results. We found that 6,7,8 were rolled most often, while 1,2,3,10,11,12 were not rolled as frequently. After we studied our results, Atticus shed some insight into why 6,7,8 may be rolled more often. Atticus proposed that 6,7,8 had more combinations for sums than the other numbers. We tested out his results, and we found out he was right! The sum of 7 has six possible combinations. You're more likely to roll a sum of 7 than any other sum. Isn't that interesting?

Our last probability activity for the week involved pulling colored cubes from a bag. The students already knew that if you put 2 blues and 2 whites into a bag and pulled one, you would have a 50-50 chance. So...we posed a new problem. What if you pulled 2 cubes at once? What combinations and chances would you pull? Most students' predictions followed one of the following:

Blue-Blue - 25%
White-White - 25%
Blue-White -50%

OR...

Blue-Blue - 33%
White-White - 33%
Blue-White - 33%

After conducting the exploration, we found out all our predictions were wrong! Our results were close to this:

Blue-Blue - 15%
White-White - 15%
Blue-White - 70%

We started thinking about the possible combinations, and we found that there was only one combination for Blue-Blue, one combination for White-White, but 4 combinations for Blue-White. Therefore, 4 of the possible 6 combinations would produce a Blue-White - which is 66% of the time. WOW!

But we didn't end there. We posed a new problem...What would happen if we put 3 blues and 1 white in the bag? We quickly predicted that we would be more likely to pull Blue-Blue than Blue-White. That would make sense, right? But our results of the experiment proved our prediction wrong once again! Our results gave us an equally likely chance to pull Blue-Blue as Blue-White. Can you figure out why? (Hint: Think about all the possible combinations.) I challenge you to pose NEW problems to this exploration and see what you can find out. Have fun!