I posted previously about a year ago an idea for using CAS in a statistics course with probability. I’ve finally had an opportunity to use it with students in my senior one-semester statistics course over the last few weeks, so I thought I’d share some refinements. To demonstrate the mathematics, I’ll use the following problem situation.
Assume in a given country that women represent 40% of the total work force. A company in that country has 10 employees, only 2 of which are women.
1) What is the probability that by pure chance a 10-employee company in that country might employ exactly 2 women?
2) What is the probability that by pure chance a 10-employee company in that country might employ 2 or fewer women?
Over a decade ago, I used binomial probability situations like this as an application of polynomial expansions, tapping Pascal’s Triangle and combinatorics to find the number of ways a group of exactly 2 women can appear in a total group size of 10. Historically, I encouraged students to approach this problem by defining m=men and w=women and expand where the exponent was the number of employees, or more generally, the number of trials. Because question 1 asks about the probability of exactly 2 women, I was interested in the specific term in the binomial expansion that contained . Whether you use Pascal’s Triangle or combinations, that term is . Substituting in given percentages of women and men in the workforce, and , answers the first question. I used a TI-nSpire to determine that there is a 12.1% chance of this.
That was 10-20 years ago and I hadn’t taught a statistics course in a very long time. I suspect most statistics classes using TI-nSpires (CAS or non-CAS) today use the binompdf command to get this probability.
The slight differences in the input parameters determine whether you get the probability of the single event or the probabilities for all of the events in the entire sample space. The challenge for the latter is remembering that the order of the probabilities starts at 0 occurrences of the event whose probability is defined by the second parameter. Counting over carefully from the correct end of the sequence gives the desired probability.
With my exploration of CAS in the classroom over the past decade, I saw this problem very differently when I posted last year. The binompdf command works well, but you need to remember what the outputs mean. The earlier algebra does this, but it is clearly more cumbersome. Together, all of this screams (IMO) for a CAS. A CAS could enable me to see the number of ways each event in the sample space could occur. The TI-nSpire CAS‘s output using an expand command follows.
The cool part is that all 11 terms in this expansion appear simultaneously. It would be nice if I could see all of the terms at once, but a little scrolling leads to the highlighted term which could then be evaluated using a substitute command.
The insight from my previous post was that when expanding binomials, any coefficients of the individual terms “received” the same exponents as the individual variables in the expansion. With that in mind, I repeated the expansion.
The resulting polynomial now shows all the possible combinations of men and women, but now each coefficient is the probability of its corresponding event. In other words, in a single command this approach defines the entire probability distribution! The highlighted portion above shows the answer to question 1 in a single step.
Last week one of my students reminded me that TI-nSpire CAS variables need not be restricted to a single character. Some didn’t like the extra typing, but others really liked the fully descriptive output.
This gets the answer quickly, but suffers somewhat from the lack of descriptives noted earlier. Some of my students this year preferred to copy the binomial expansion terms from the CAS expand command results above, delete the variable terms, and sum the results. Then one suggested a cool way around the somewhat cumbersome algebra would be to substitute 1s for both variables.
CONCLUSION: I’ve loved the way my students have developed a very organic understanding of binomial probabilities over this last unit. They are using technology as a scaffold to support cumbersome, repetitive computations and have enhanced in a few directions my initial presentations of optional ways to incorporate CAS. This is technology serving its appropriate role as a supporter of student learning.
OTHER CAS: I focused on the TI-nSpire CAS for the examples above because that is the technology is my students have. Obviously any CAS system would do. For a free, Web-based CAS system, I always investigate what Wolfram Alpha has to offer. Surprisingly, it didn’t deal well with the expanded variable names in . Perhaps I could have used a syntax variation, but what to do wasn’t intuitive, so I simplified the variables here to get
Huge Pro: The entire probability distribution with its descriptors is shown.
Very minor Con: Variables aren’t as fully readable as with the fully expanded variables on the nSpire CAS.