I ask students to define probability, and, give an example of a probability that makes sense to them. I emphasize the "makes sense" component since so much in probability feels groundless to students. So, I am looking to see what kinds of statements they are comfortable making in probability.
I like students to share if they have a very basic approach, something like "we have 4 marbles in a bag and one is green. So, the chance of picking a green is 1 out of 4." Here, I want to remind them that they might express this as a decimal or percent. I also stress that this example captures a fundamental idea in basic probability. The 1 out of 4 ratio reminds us that probability is simply a ratio between "what you want to get" and "how many options there are." It is important for students to understand the first number is always a reflection of what we want to get and the number or ways to get that thing. With the green marble example there is only 1 way to grab the green marble. Namely, to reach in and grab that marble.
Students will then ask questions like, "well what if we want to cut the bottom out from the bag and let the marbles fall into my hand?" This is in fact a great question. I ask students to think about the idea that the assumption is "we are going to get that green marble by reaching into the bag and pulling it out." We could consider other cases, but that changes the nature of the problem.
To demonstrate the importance of all these distinctions, I give students a packet of about 8 probability problems: Basic Probability 1. We review these problems in the summary.
As students work on Basic Probability 1, I gather their questions and ideas for the summary. If students finish the 8 questions early, I ask them to write their answers as percents (spiraling back to their percent work from past years). My main goal is to gather misconceptions and unique strategies for the class discussion, but I like to also make up questions that relate to their worksheet. I try and push their thinking by asking deeper questions like, "why might we need empirical probability?" I want students to understand that empirical probability informs of as to whether our theoretical instincts are correct. This is analogous to the scientific method and their experience testing out hypothesis.
Each of the 8 problems brings out different questions (and I respond to each in the summary). One question I love to discuss is question 4. In this question, students are given a table and asked, "what is the empirical probability that the spinner will land on a prime number in the next spin?" I like this question for a few reasons. First, it helps me reinforce the phrase "empirical probability." Secondly, it helps me remind students that 1 is not a prime number.
"Well what is it?" they ask.
I remind them that 1 and 0 are very special numbers and it helps to think of them as the multiplicative and additive.
The other great part of this question is the phrase, "what is the empirical probability that the spinner will land on a prime number the next time?" A misconception here is over the phrase "the next time." Many students take add or subtract more events and somehow change the probabilities. I find that it is here that students are learning about the concept of having in place of friends and family.