Lesson 4.4 – Conditional Probability and Independence
- Find an interpret conditional probability using two-way tables.
- Use the conditional probability formula to calculate probabilities.
- Determine whether or not two events are independent.
We previewed the Activity with a short discussion about what it meant for two events to be independent. We used the context of coin tosses to understand independent events as knowing whether or not one event has occurred does not change the probability that the other event will occur. We used the context of snow days to talk about dependent events (knowing whether or not Thursday is a snow day changes the probability that Friday is a snow day).
Activity: Do you prefer English or Math?
We started by having all the students come to the front white board to put a tally mark in the appropriate location. Then we counted up the tally marks. From there, students worked in pairs on the rest of the Activity. We made the suggestion to students to write all probabilities as fractions.
Most students will be able to calculate the probabilities in questions #4 and #5 without any formal introduction to the idea and notation of conditional probability. We introduced the concept and notation of conditional probability when we were going over the answers.
Students struggled a bit with question #6. We asked them “if I know whether or not a person is female does that change the probability that they prefer English?” We wanted them to compare P(prefers English | female) to P(prefers English | male).
In an effort to really develop a good understanding of conditional probabilities, we did not introduce the formula for conditional probability. We really wanted students to be able to read a question and identify the “condition” that is being given. This given condition tells us which part of the two way table (or Venn Diagr
am) we should use to calculate a probability. To help students with this idea, we had them use index cards to cover up the part of the two-way table (or Venn Diagram) that is not needed, leaving only the information from the given condition (see pictures below).