General Addition Rules
Chapter 5 - Day 4 - Lesson 5.2
Apply the general addition rule to calculate probabilities.
Use a two-way table or Venn diagram to model a chance process and calculate probabilities involving two events.
What is the probability of being a brown eyed female?
Before starting this activity, you might want to start with a discussion about the word “OR”. Unfortunately, in English “OR” has two possible meanings (Applebees vs. Starbucks “OR”). In mathematics, we want the inclusive Starbucks “OR”.
For the activity, you will need to choose 10 students from the class. Try to pick exactly one student who is male with blue eyes and exactly one student who is female with brown eyes. You will see why in just a second here.
Create a two-way table on the front white board with the two variables “Gender” and “eye color”. Have the 10 students put tallies into the table and then total them up. Students will need this data to complete the activity.
The biggest idea here is the General Addition Rule and the goal is to have students discover this idea rather than handing them a formula. Keep the 10 students at the front of the room as you work through each of the probabilities in the activity. Make the standing students raise their hand and step forward when the event describes them.
Here is the critical point in the lesson: #4 P(male OR blue eyes).
Have all the males step forward and raise their hand. Count them. Suppose P(male) = 5/10
Have all the blue eyes step forward and raise their hand. Count them. Suppose P(blue eyes) = 3/10
We should be able to simply add these two probabilities to get P(male OR blue eyes), so 8/10
Have all the males or blue eyes step forward and raise their hand. Count them. Why are there only 7? We were supposed to get 8. Students will realize that someone got double counted (like Turner). If we subtract out the overlap, we arrive at the right answer.
Why didn’t we have this problem when we found P(blue eyes or brown eyes)? Because those events are mutually exclusive (no Scherzer allowed here). But for P(male OR blue Eyes) the events were not mutually exclusive, so we ended up double counting the overlap, and it must be subtracted out. This is the General Addition Rule.