After thinking about ways to make our learning and practice with binomial probability interesting and meaningful to my students and falling short each time I remembered an example my professor used in class. We used binomial probability to determine the probability that a basketball player would make 8 free throws given 10 attempts with a shooting percentage of 80% (I never forgot this as it reminded me of my years playing high school basketball and always having to shoot 10 free throws before we left the gym). Knowing I did not have time to get my students to the gym to shoot free throws I came up with a modified version using corn hole. For my fellow math teachers you might want to scroll down to pass the explanation of binomial probability, for everyone else, I promise it is not has hard as it may sound.
How do we use binomial probability?
Binomial probability is used to find the probability of a binomial experiment. A binomial experiment must have the following characteristics:
- The experiment consists of n repeated trials. – In our basketball scenario this would be the 10 shots that the player gets to shoot.
- Each trial as 2 outcome (called success and failure) – This is obvious for our scenario as making a basket is successful and missing a basket is a failure. This can also be applied to tossing a coin or taking a multiple choice test. You just need to define your success and your failure.
- The probability (P) for success is the same for each trial – The probability that our basketball player makes shot is 80% every time he takes a shot.
- The trials are independent – The outcome of one trial does not affect another trial. If the player makes the first shot it does not affect the probability of him making the next shot, he still has an 80% probability.
We then use our graphing calculators to find the probability for the different scenarios. I will now go into how we used binomial probability and corn hole.
Cornhole to Collect Data Probability
With the corn hole board set up and ready to go in the classroom I asked for any volunteers that wanted to toss the bags. Each student that volunteer was given 8 bags to toss. For the purpose of our lesson we had the following outcomes: score points, don’t score points. This means that if the player tosses the bag on the board or in the hole it is counted as a success (it is not perfect, but we simplified our experiment to use binomial probability). As the students took turns we calculated the probability for each student that when they toss a bag they will score points. This is a simple calculation: # bags score points/# bags tossed. All of the students recorded (on their whiteboard desks!) the probability for all of our volunteers.
Using the Binomial Probability
After all of the volunteers went through I gave the students different scenarios to calculate. The first scenario was given 10 attempts what is the probability that the person tosses exactly 6 bags that score points. Since we had 8 volunteers the students had 8 different probabilities to calculate. After the students calculated the probabilities we discussed their findings. I was lucky enough to have 1 student have a probability of scoring as 100% which led to some great discussions! Many students were shocked that this students had a 0% probability of making exactly 6 when he had the highest probability of scoring. We discussed how he does not miss and if he only made 6 of 10 it means he would have missed, but he doesn’t miss, so the probability is 0.
We continued with different scenarios that included at least 8 of 10, at most 5 of 12 and so on. After each scenario we discussed our findings. For one of our scenarios a student had an 85% probability of meeting the goal so I had the student come back up to see if it would happen. Now I think the pressure may have gotten to her, but she was one bag short of the goal. Again, great discussion as just because their is a high probability that something will occur does not mean it is a guarantee that it will occur. In our case, we witness 1 of the 15 times out of 100 that this student would not meet the goal.
This lesson was a great way to get students involved in the material and bring about a discussion for the different scenarios. With 8 different volunteers the students had plenty of practice using their calculators to find the probabilities. When I do this again I will have more students come back up to toss again. The new whiteboard desks (in the process of writing that post) helped the students organize their work and share their work with one another. We also had 2 visitors to our room that day and the desks really helped my students show our visitors what they were doing instead of working on paper.
They also wanted to see if I had any corn hole skills…let’s just say I was a little rusty.