I like to briefly revisit the Monty Hall problem. I think it is the ultimate example of conditional probability. When the contestant needs to either "switch" or "stay" there are only two choices left, one a car and one a goat. It seems hard to believe that it isn't a 50% chance of winning. It seems impossible that you double your chance of winning if you switch. To remind them of the thinking behind this problem, we revisit the tree diagram:
To support the tree diagram, I show the calculations:
1/3 * 1 + 1/3 * 1
I want students to understand that we multiply independent successive events and add mutually exclusive events. In this case, a person has 2/3 of a chance of picking a goat and are certain to win if they switch. If we think of the goats as separate, goat A and goat B, we can ask what is the chance of picking goat A and then switching to win? Here you have a 1/3 chance of picking goat A and then are certain to win. The same is true for Goat B. Since you either choose Goat A or Goat B, we add their probabilities. This centers on the meaning of mutually exclusive.
I also like to ask, "what is the chance you will pick a goat or a car?" Here using the key word "or" students realize that the answer is 100%, since you have a 2/3 chance of picking a goat and a 1/3 chance of picking a car. The idea is that you are bound to pick something.
The Monty Hall problem gets them thinking about the context of conditional probability, but I like to solidify their work through a more rigid example:
This problem spirals back to empirical probability and pushes student thinking around conditional probability. For example, when they answer, "what is the probability that a 7th grader will like ice skating?" Students need to focus in on 7th grade and change the total number of students they analyze. It is no longer about all the students in the school, but only about 7th graders. For that reason our denominator is the total number of 7th graders, not the total number of students in the school .
I hand out the worksheet for this part of class as students finish the problems from the start up:
I like to check and see that my students were able to answer the introduction question before they get to far into the worksheet. As always I am looking for misconceptions and successful strategies to share in the summary. These problems are tough, but students find very clever ways of dealing with the problems (see Summary for more about this).
Dealing with conditional probability presents many challenges for students. One place to start in the summary is to revisit the importance of the phrases "with replacement" and "without replacement." I like to get a transparent cup and place some type of colored item inside, like marbles. Then I like to simulate the significance of replacing or not replacing an item in a series of trials.
For example, if I have 2 blue marbles and 1 red marble, the chance of picking 2 blues with replacement is 2/3 * 2/3 and the chance of picking 2 blues without replacement is 2/3 * 1/2. The idea is that not having replacement may or may not significantly alter your chances in any situation. If students need more support, we refer to the last question on the template from the partner work portion of class.
After answering various questions from the class work, I save the best for last:
Although I always took a formulaic approach to this problem, many students were able to solve it by thinking backwards.
They know that the chance of it snowing on both days is 3/10 and that one day has a chance of snowing at 3/5. They recognize that in order to get a probability of 3/10 from two independent events that those two events need be multiplied to result in 3/10.
In other words, 3/5 times what gives 3/10?
This brings up fundamental algebraic thinking and some basic balancing of equations. It is a fun way to end the class (by reviewing the first problem).