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I like introducing probability by discussing the Boy or Girl or Bertrand paradox.
What other (short) problem/game provides a motivating introduction to probability? (One answer per response, please)
P.S. This is about a gentle introduction to probability, but in my opinion it is relevant for statistics teaching as it allows to further discuss about discrete events, Bayes's theorem, probabilistic/measurable space, etc.
A good example of showing how people are non-random is to get the class to write down a number between 1 and 10. You then ask the 1's, 2's, .. to stand up.
What happens is that the majority of the class choose 7 and very few choose 1 and 10. This leads on to interesting questions, such as:
A standard example is the Monty-Hall game.
Here's how I approach this example:
I really like any problem that has some result that is counter-intuitive to what we'd like to think. The problems thus far are classics in the field of probability, so I'll add my favorite classic problem: The Birthday Problem. I always found it amazing that there was such a high probability of having two people with the same birthday with such a small sample.
At risk of sounding too simplistic, I think the best problem to introduce depends on who you are talking to.
For example my arts friends freak out when I talk about math and stats, but then I tell them they shouldn't be afraid because they speak math all the time. So I give them examples such as "What are the odds it will rain today?", you don't acknowledge you're doing the computation but you are assessing some probability in your mind. So for them I like to pick very relateable problems dealing with weather and emotions ("For example, given you are depressed, how likely is it do be raining outside?") and show them the math behind how we might answer that. Then later after they have discovered an intuition for mathematical problem solving I tell them what the terminology is for it. AND yes I have gotten my arts friends to sit willing through that!
I personally learned statistics better when I had a problem in my domain I understood very well. I find when you understand a problem very well it becomes easier to understand the math. I think too often people just learn by rote and look to fit problems they've already seen onto new ones rather than trying to understand each problem.
The Drunkard's Walk by Leonard Mlodinow is full of such examples, including one on the meaning of a Positive HIV test that is 99.9% accurate. Using bayesian statistics, the actual odds of a positive test are less than 10% (a similar example is detailed in chapter two of Agresti's Introduction to Categorical Data Analysis book). Another example (i break the one example per answer but this is essentially the same problem from conditional probability) is from the Simpson trial, where one of Simpson’s lawyers, Alan Dershowitz, noted that even though Simpson beat his wife, that hardly mattered, because in the United States, four million women are battered every year by their male partners, yet only one in 2,500 is ultimately murdered by her partner (1 in 1000), so, by the 'reasonable doubt' criterion, this is irrelevant. The jury found that argument persuasive, but it’s spurious. The relevant question was what percentage of all battered women who are murdered are killed by their abusers, which ain't 1 in 1000, but rather 9 in 10.
For a gentle introduction, I like examples using 2x2 contingency tables. The diagnostic testing example as mentioned above, where the Probability of a positive test result given disease is not equal to the Probability of disease given a positive test result. Also, one can use designs with different sampling schemes, such as the cohort study vs. the case-control study, to illustrate how that affects what probabilities can be estimated.