Five probability problems to help us think better

Probabilities are a curious type of numbers. If properly understood, they can help us see the world around us in an appropriate context. Sometimes these numbers can be counterintuitive and downright confusing and sometimes misleading. This article from The Atlantic presents 5 classic problems in probability that can help us think better in probability and statistics. Some of these problems confounded experts. It pays for anyone to pay attention to these problems.

These 5 problems are the Monty Hall problem, the birthday paradox, Gambler’s ruin, Abraham Wald’s memo and Simpson’s paradox.

One common characteristic to all these problems is that they are in some sense paradoxical. The facts on the surface can lead us down one path to a wrong solution. On the other hand, the correct result can be so counterintuitive that it seems absurd.

Take the Monty Hall problem for example. When it appeared in a column authored by Marilyn vos Savant in Parade Magazine in 1990, it drew a great deal of angry responses from readers, some of whom were holders of PhDs on math and statistics (they said so in their disparaging responses). These experts in math and statistics all claimed that the solution proposed by vos Savant was wrong and she should know better. Some of these angry remarks are repeated here.

These experts were wrong! It turned out that even experts can be confounded by probability numbers too. As a result of the controversy, the Monty Hall problem is a probability problem that is known widely and is covered in most standard introductory texts on probability and statistics. The Monty Hall Problem: The Remarkable Story of Math’s Most Contentious Brainteaser, a book entirely on the subject of Monty Hall problem, is authored by Jason Rosenhouse (Oxford University Press, 2009).

Refer to the above link for a quick introduction to these 5 problems. Three of the problems have been discussed in several math blogs affiliated with this blog. The following are the links to these blog posts.

The following blog posts discuss other classic problems in probability.


The Story of Billy Barr

I recently came across an article in The Atlantic that tells of a remarkable man named Billy Barr. His claim to fame is his outsize impact on climate science. Thanks to his effort, scientists now have a deeper understanding of the effect of climate changes on the Rocky Mountains and other similar alpine environments. Yet he is not a scientist. He certainly did not set out to go to the Rocky Mountains to become a scientist. The article in The Atlantic is about his remarkable life story. Here’s a video from the National Geographic about Billy Barr.

In the last couple of decades, small but perceptible changes in the high alpine environments had caught the attention of climate scientists. For example, spring snow seemed to melt a little earlier. The flowers blossomed a little sooner. However, they could not make much sense of the changes without a historical context. Without historical data, scientists would not know whether the recently observed patterns were due to random fluctuations or actually represented a clear break from the past.

Barr had made Gothic Mountain in Colorado his home in 1973. Billy Barr did not move from the East Coast to the Rocky Mountains to become a scientist. He went there to find inner peace. He started data recording in his first year there partly as a way to combat boredom and partly to have a point of reference for future winters. He measured snow levels, animal tracks, and in springs the waking of hibernating animals and first joyful calls of bird returning. He filled one notebook, then another and has been doing so continuously for 44 years!

Luckily for climate science, Billy Barr lives very close to the Rocky Mountain Biological Laboratory (RMBL). He became a volunteer there early on in his time in the mountain as a caretaker and later keeping track of expenses as an accountant. Amazingly the scientists did not know about his data collection habit until late in the 1990s!

Once the scientists at RMBL found out, they realized Billy Barr’s data was a priceless trove of data that could help them make sense of the warming trends in the world. Billy Barr’s data are the basis for dozens of research papers on climate science. His data on the high alpine environment has provided an unexpected glimpse into a prior world that scientists never recorded. His notebooks were filled with data on first and last snow, the snowpack levels in between, and when hibernating animals wake and when the birds return after winters.

What is the problem with earlier snow melt and earlier flower blossom? The answer is that such seemingly small changes result in seasonal imbalance in the alpine environment (also called phonological mismatch). A concrete example is the broad-tailed hummingbird. The hummingbird relies on nectar from the glacier lily. Barr had tracked the hummingbird’s return each spring and the first blossoms of the glacier lily. In the past the return of the hummingbird and the blossom of glacier lily were in sync. The glacier lily now flowers 17 days earlier than 40 years ago. If the same warming trends continue, it is likely the broad-tailed hummingbird will completely miss the nectar of the glacier lily, thus spelling the doom for the bird. What happen to the broad-tailed hummingbird will set off a negative chain reaction for butterflies, bees, hibernating mammals, and the other animals that depend on them.

The example of the broad-tailed hummingbird suggests that the ecosystem in the alpine environments might be rapidly approaching a tipping where a small change in the system can result in a drastic change overall.

The snowpack in the Rocky Mountains quenches the thirst in the cities in the surrounding regions. Forty million people rely on the Colorado River for water. Not surprisingly, Barr’s data helps shape water policy for Southwestern region of the United States. For example, hydrologist Rosemary Carroll used Barr’s snowpack data and other sources to model groundwater flows to the Colorado River.

Billy Barr is a fascinating story. For a fuller story, read the Atlantic article or view the National Geographic video. Or you can Google Billy Barr.