# What is your favorite problem for an introduction to probability?

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.

## 6 Answers

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:

• How should you choose a random number.
• Designing an experiment?
• What do we mean by random?
• Is there an explanation for the appearance of 7? – user28 Sep 27 '10 at 18:52
• My general hand-waving explanation is this: people avoid {1, 5, 10} because they are too obvious and therefore "not random". Numbers less than 5 - well who wants a small RN! People then tend to go for the middle number between 5 and 10. I've tried this example six times now (in classes of size ~100) and it's worked each time. – csgillespie Sep 27 '10 at 21:49
• And of course, 17 is the least random number. catb.org/~esr/jargon/html/R/random-numbers.html but my favorite random number is 37: jtauber.com/blog/2004/07/09/… (though, also see scienceblogs.com/cognitivedaily/2007/02/… ) – ars Sep 27 '10 at 23:17
• I think this showing that "randomness" cannot be fully defined. If you start defining "randomness" to much, then it becomes systematic. One good example is shuffling cards - if you do it in a systematic way, then the shuffling achieves nothing. – probabilityislogic May 21 '11 at 0:50

A standard example is the Monty-Hall game.

Here's how I approach this example:

• Give the class sets of three cards and get them to play the game in pairs.
• Each pair plays the game following a particular strategy, i.e. always switching doors.
• Afterwards, I use the number of times the class won to calculate a Monte-Carlo estimate of winning.

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.

• I agree with you and about a decade ago collected a bunch of such problems for a course (see quantdec.com/envstats/homework/class_03/paradox.htm ). However, there is a strong pedagogical counter-argument: Probability itself can be confusing, so if you start off with counter-intuitive examples, you risk losing your audience forever (like Augustus DeMorgan, a pioneering probabilist, who later in life completely gave up on probability as hopelessly difficult!). So caution is in order here, especially if you want to motivate people in an introductory setting. – whuber Sep 27 '10 at 15:32
• I think it causes polarization. The students who are not interested in mathematics/probability will become confused, and the inquisitive/interested students will be inspired to learn more. Like you said, it might be best to exercise caution. Nothing could be worse than a confusing teacher presenting a confusing example! – Christopher Aden Sep 27 '10 at 17:12

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.

• This is my favorite example too (HIV test), but unsure if conditional probability is too "advanced" given the introductory nature (plenty of studies showing that it's not too intuitive). If you do teach this, I recommend perusing Gigerenzer and the frequency method: library.mpib-berlin.mpg.de/ft/gg/GG_How_1995.pdf – ars Sep 27 '10 at 23:21
• @ars:> maybe first you state them all the relevant informations in table form, then the problem "what do you think is p(AIDS|test=1)?", then the counter intuitive punchline, only then you show them the problem re-casted as a 'tree' (where the final 4 nodes are all possible cases) and the branches show the respective probability. In my experience, the last leg need not be understood by everybody, but it has to convey the importance of having a principled way of thinking about these issues. – user603 Sep 28 '10 at 0:08

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.