14.2 Trickier probability problems

The thing that makes probability problems the darling of math contest writers everywhere, is also the thing that makes them frustrating for so many students: no two problems are exactly alike. Most probability problems can be solved with the multiplication rule, combined with a lot of good, hard thinking about the problem.

I’m going to present two scenarios with five questions here, in the lesson plan. The idea is for you to talk them through with the class. In each case, explain the scenario and the question clearly. Then give them a minute or two, with no guidance, to think about it. Then take their answers and go over the correct answer very slowly and clearly. None of them should be presented as if it were a symbol of a whole, unique, important class of problems. Each should be presented as simply another example of you can solve a wide variety of problems, if you’re willing to think about them patiently and clearly.

The right way to think about this problem is as the reverse, the “not,” of the previous problem. We said that 25 out of 36 times, neither die will roll a 1. So the remaining 11 out of 36 times, at least one of them will. This is an example of the “not” rule we got from last night’s homework: the probability of “no ones” is $\frac{25}{36}$ , so the probability of NOT “no ones” is $1-\frac{25}{36}=\frac{11}{36}$ . (*It’s interesting to note that the “naïve” guess of $\frac{1}{3}$ is not too far off, and makes a reasonable approximation. If you have a 1 in 10 chance of doing something, and you try three times, there is a roughly $\frac{3}{10}$ chance that you will succeed at least once—but not exactly $\frac{3}{10}$ .)

The last thing you need to assure the class, before you hit them with the worksheet, is that no one is born knowing how to do this. Probability problems are just like everything else: they make more sense, and get easier, with practice. It’s OK to get frustrated, but don’t give up!

Then give them the worksheet. Ideally they should be able to make a good (10-15 minute) start in class, and then finish it up for homework. Expect to spend a lot of the next day going over these. It’s worth it.

Homework