We’re approaching the chapters in the text on quantum mechanics so today we brushed up on probability. I found this amazing paper online that included a full tutorial and activities to do with a class. The name of the paper is “Laboratory -Tutorial activities for teaching probability.” by M.C. Wittmann, J.T. Morgan and R.E. Feely at the University of Maine. The tutorial starts with flipping coins and looking at the probability of getting heads verus tails for 1 penny flipped 10 times, and then flipping three pennies at a time and comparing the different outcomes. Then we moved on to rolling two dice and looking at the probability of their sum being a certain number. For example, there is only one way to get ‘snake eyes’, a sum of 2, or a sum of 12. But there are quite a few rolls that add up to 7. The students each rolled two dice 8 times and made a histogram of their results and then we added up the data for the whole class (photo on left) and saw how that compared more favorably to the predicted probability histogram. We discussed how a larger set of data more closely resembles the predicted curves. Another important point that we discussed was that the probability of any particular roll, did not depend on what had happened on the previous roll. Just like when you flip a coin 4 times and even if you got 4 heads in a row, the chance of getting a head on the 5th flip, is still 50%, its not any higher or lower just because you already got 4 heads.
Then we threw some physics into the problem. I drew a cloud on the top of the marker board and single rain drop falling from it. I divided the board below the drop into 3 separate sections, one above the other, and asked the students, if I was to randomly take a photo of the raindrop, is it more likely to be in one section or another, or is it equally likely to be in any of the 3 sections. Some students will think its equally likely to be found any of the 3 spaces since they are the same size, but some remembered that the raindrop will be accelerating and going faster and faster as it drops, therefore spending more TIME in the top section and the least amount of time in the bottom space. So you are more likely to find the raindrop in the top region.
We then took this a step further and used the Video Physics app on an iPad to record the motion of a ball being thrown up in the air and coming back down. Just like the raindrop, we divide the vertical space into three sections and find the probability of the ball being in the top, middle or bottom space. Here is a photo with the ball locations already marked. Students took these printouts and used a ruler to divide the space equally and then counted the number of frames the ball was in each section. The probability of the ball being in the top 3rd of the throw was equal to 56% (14 frames out of a possible 25).
Finally, we took a look at the glider on the airtrack. Springs were attached between each end of the cart and the ends of the airtrack so the cart would oscillate back and forth. Again, we used the iPad to record the motion and this time we divided the region into 5 sections. This histogram looks a bit different then the ones we’ve done so far since the cart is moving slower at the extremes of its motion as it turns around, its more likely to be found in region A and E, and less likely to be found in region C where its moving the most quickly.
What does this have to do with quantum mechanics? In quantum mechanics we don’t know exactly where things are, but we can determine the probability density of finding a particle, like an electron, at a particular location. Similarily, today we found the likelihood of finding the cart in a particular location.
Now I’m going to go to the University of Maine’s website and see if they have more tutorials because this was a great set of activities.