Today I dove in a little more to see if he could see some of the patterns that emerge in the distribution. We started with a quick review and a look at data from a few simulations I ran:
Next we looked at the data from four simulations with an averages of 1, 2, 3, and 4 events expected per year. It was a little hard for him to see the overall pattern, but after a few hints he was able to see what was going on:
To wrap up today, we looked at the pattern from the simulations and tried to write down the pattern that we’d expect to see for an event that happens 5 times per year on average. At the end of this video he was able to write down a formula for the general pattern!
For our project today, I though it would be fun to talk about the Poisson distribution. For me it is one of the most interesting and important ideas in probability. This question, for instance, is fascinating -> If a random event happens on average once per time period, what is the probability that it happens twice?
I started the introduction with a version of the idea I mentioned above and asked my son for some estimates of what he thought the answer would be:
Then we looked at some simulations. Here I’m looking at the idea of a random event that happens on average once per year and chopping the year up into 52 weeks:
Next I chopped the year up into 365 days – would we get different answers?
This project turned out to be a little more interesting to my son than I was expecting – I’m looking forward to exploring Poisson distributions a bit more next week.
Last week I learned about an really interesting probability problem from Pasquale Cirillo:
Today I asked him to think about the problem while I was out and when I got back home he walked me through his solution:
The last video shows his general approach – now he calculated the asnwer:
To wrap up I showed him how to modify his original argument just a bit to avoid the infinite series calculation. This is a much shorter way to solve the problem, but does require a bit more mathematical sophistication:
I really love the problem and think that it is a great one to share with kids. Even if kids can’t quite solve it, it would be really fun to hear their thought process and how they might estimate the probability of winning.
Yesterday we did not get to the optimal solution, but rather looked at the strategy of stopping when you get a 6 on the first or second roll, and then at stopping when you get a 4 or higher on the first or second roll. I asked my son to think about the problem a bit more this morning while I was out and he was able to find the optimal solution.
Here’s what he did in his own words:
Next he showed how he used Mathematica to help him find the best solution:
Finally, I showed him an alternation approach to finding the optimal solution that comes from working backwards. This is the approach that Cirillo takes in his discussion of the problem: