Analyzing Binomial Random Variables
Day 67 - Lesson 5.4
Calculate and interpret the mean and standard deviation of a binomial distribution.
Find probabilities involving several values of a binomial random variable.
Use technology to calculate binomial probabilities.
Today’s lesson is an extension of binomial distributions from yesterday. Really the only new content is calculating a mean and standard deviation. The rest of the learning targets are an extension of previous learning in probability.
Our context today is a soccer shootout where the soccer team is going to kick 5 penalty kicks. We verified that this is indeed binomial using BINS. We wanted students to see the connection between the new formula for finding a mean, n × p, with the previous formula, mean = ∑ x × p. To do this we had students create the probability distribution. In their groups they completed a table showing the various probabilities of making 0 through 5 goals. They calculated the mean using ∑ x × p. After doing this, a few students noticed that this was the same as n × p, which of course is our new formula! We asked them to share out their thinking with the class, and we all agreed that they had sound logic so we added in to the margin that mean = n × p. Make sure to explain that this is only true for a binomial distribution. Think of it as a short cut, they can still use ∑ x × p if they forget.
You will need to tell students the formula for standard deviation. We couldn’t come up with a clever way of discovering it, and that’s ok. The rest of the activity focuses on using the probability distribution to find probabilities. Students should remember that “OR” means add and P(At least 1) = 1 – P(None). We showed the class at the end how they could use the applet but we did not use it for the activity. We wanted them to do the work without it. It will be needed for the application problem. We entered the information into the applet for them and projected the results.
Overall this activity went really well. It’s just far enough out of the students reach that they had to think deeply but still within grasp so they weren’t intimidated.
Try to use the same language for the mean and standard deviation that you have been using all year for the interpretations. For example, we always emphasize the “typically varies” part of standard deviation.