I’m teaching part of a basic medical statistics class this summer. It’s been about a decade since I’ve taught basic probability and statistics and I now have different ideas about what is important. For example, I now think it’s more important that a beginning class understand the law of small numbers than the law of large numbers.
One reason for my change of heart is that over the intervening years I’ve talked with people who have had a class like the one I’m teaching now and I have some idea what they got out of it. They might summarize their course as follows.
First we did probability. You know, coin flips and poker hands. Then we did statistics. That’s where you look up these numbers in tables and if the number is small enough then what you’re trying to prove is true, and otherwise it’s false.
Too many people get through a course in probability and statistics without understanding what probability has to do with statistics. I think we’d be better off “covering” far less material but trying to insure that students really grok two or three big ideas by the time they leave.
I’m a confirming data point in your change of heart. I read this post quickly, then went back and read the “typical summary” for probability and statistics — pretty much sums up my experience in the course back in college. I laugh now, but…
A couple of key questions: (1) Who are your students? and (2) How will you evaluate them?
My feeling is that conventional statistics courses have tended to emphasize formulas and tables of critical values, which is not very useful at all. I think students need to learn some basic principles of statistics.
But it’s also useful for them to be introduced to standard stuff about parametric and nonparametric tests for independent groups as well as paired data and for continuous data as well as dichotomous data. Plus correlation and regression.
Diagnostic tests play an important role in medicine, and they correspond closely to hypothesis testing ideas.
And of course, confidence intervals need to be emphasized.
I’ve stopped teaching the sampling distribution of the sample mean (search on blog for details).
As a math teacher myself I am often astounded when students don’t understand the basics of what a percentage is … the amount divided by the total is equal to the number out of 100 … which is the proportion.
Combinatorics (permutations and combinations) are related, in that the amount of ways something can happen gets divided by the amount of total outcomes.
Students and teachers will take mental shortcuts, such as … “and” means multiply, “or” means add … but these lead to misconceptions.
I do a thought question before teaching prob stat. Consider if you were picking from 2 groups: Beverages (H20, tea, soda, coffee) and snacks (chips, cookie, peanuts). What does picking a Beverage or a Snack mean? 4+3 options. What does picking a Beverage and a Snack mean? 4 times 3 = 12 options.
Then do the same questions but include overlapping groups. Say Black cards is group A, and Face cards is group B. What does picking a Black card or a Face Card mean? (Point out that some face cards are also Black). 26 + 12 – 6 = 32 options.
Now for the important question. What does picking a Black card and a Face Card mean? Does it mean 2 cards, or one card that is both Black and a Face card? Is there an implied order in the assumption that we are picking two card: such as is the Black card first, and the Face card second?
Also, if we are really picking from one deck of cards, these are really not 2 different groups, but two different characteristics within the one group (the deck of cards). The original process of picking from Beverages and Snacks were however 2 separate groups.
These are all small differences, but very important when it comes to understanding how we are counting.
Assuming we are picking 2 cards from a deck out of 52, if we deal with the cards groups by dividing an amount by a total, then we can determine the total number of outcomes by multiplying 52 times 51 . The amount would be counting the event that we are interested in.
Now returning back to calculating probability. If we want P(Black or Face) the amount of Black or Face cards = 26+12-6 = 32 , so the amount picking two would be 32 times 51(because you don’t care about the 2nd card) , and the probability of picking a Black or a Face card in two picks is ….. (32*51)/(52*51)
Which is the same as (32/52).
That of course assumes, we don’t care about the 2nd pick. What if we do care?
What if we want the two cards to be either a Black or a Face card? We just multiply 32*31 (remember 32 is the total amount of Black or Face cards, minus Black Face cards) and divide by 52*51.
Which is the same as (32/52)*(31/51).
You could also show this with a sample space array, where there are 32 rows are versus 31 columns (the one less due to the first picked card), and then the 52 rows are versus the 51 columns.
Now if we want P(Black and Face) then the amount must include the two possible permutations (Black, Face) or (Face,Black) while paying attention to the intersection (Black Face, Face Card that is Black). So the amount is “26*12 plus 12*26 minus 6*5” or 26*12*2 -6*5. To get the probability divide by 52*51