Desmos, the online graphing calculator, activity builder, and general favorite of the MTBoS does phenomenal work.
I found myself wondering over the past few days about the statistics and probability world and how there isn't a Desmos parallel in that realm for easy experimentation. You can put an algebraic expression in Desmos and graph it in a few keystrokes. You can solve a problem algebraically, and then graph to see if you are correct. There are multiple ways to confirm graphically or numerically what you have solved algebraically, or any other permutation of the three.
It's also easy to put a bunch of marbles in a jar, pick one, replace, and repeat, though this becomes tedious beyond a few trials. Such a small data set often isn't enough to really see long term patterns, particularly in cases where you are trying to test whether the theoretical probability you have calculated is correct or not. For subtle cases that involve replacement versus no replacement, the differences between the theoretical probabilities of events are small if there are enough marbles in the jar.
Creating a simulation and running it half a million times is possible in a spreadsheet or a number of computer languages, but the barrier to entry there is not trivial. I've written simulations myself of various problems and usually make predictions for what I think is going to happen. I then will usually work to find the theoretical probability by hand.
So what would this sort of probability playground look like? There are some examples out there already. Here's one from CPM for small numbers of trials. I haven't done an exhaustive search, but I haven't seen anything that truly allows full experimentation at the level I'm hoping to achieve. Here are some ideas for what I would love to see exist:
- Natural language definitions for sources of possible outcomes. By this, I mean being able to define outcomes verbally. This might mean "rain" and "no rain", with the assumption that having only two labels means these events are complementary. This might mean we define numbers of items for each possible outcome, or simply enter the probability of each as a decimal. The key thing is that I do not want to require labeling events as A or B, and throwing notation around. Let's see if we can make this as visual and easy to explore
- Ease of setting up conditional outcomes for compound events.. If event A (I know, I'm breaking the previous rule here) happens, only B and C are possible, and event D is only possible if event A does not occur.
- Sinks that easily allow for large numbers of trials. I might want to have a single trial be generated a million times - tell me the proportion of all of the different outcomes. Make it easy for me to count up instances of binomial probability and see how many times, out of ten, I get three or more successes. Tell me when I'm not looking at all of the possibilities. For example, give me some visual indication that when I'm picking two marbles from a jar, that if I only have both red or both blue in my possible outcomes, I'm missing outcomes in which there is one of each.
- Make it easy to tap into existing complex data sets for exploration purposes. Include some data sets that are timely and relevant. The US election comes to mind.
I realize also that this is a tall order, but I've seen how far the Desmos team has explored the algebraic/numerical space. Now that they have expanded into the Geometry space through their beta, I wonder if they (or someone else for that matter) has something like this probability exploration tool on their roadmap.