Sampled (Underfoot)

Some interesting statistics from the American sociologist Elizabeth Wrigley-Field:

Here are three puzzles.

• American fertility fluctuated dramatically in the decades surrounding the Second World War. Parents created the smallest families during the Great Depression, and the largest families during the postwar Baby Boom. Yet children born during the Great Depression came from larger families than those born during the Baby Boom. How can this be?

• About half of the prisoners released in any given year in the United States will end up back in prison within five years. Yet the proportion of prisoners ever released who will ever end up back in prison, over their whole lifetime, is just one third. How can this be?

• People whose cancers are caught early by random screening often live longer than those whose cancers are detected later, after they are symptomatic. Yet those same random screenings might not save any lives. How can this be?

And here is a twist: these are all the same puzzle.

• Answers here: Length-Biased Sampling by Elizabeth Wrigley-Field

Proxi-Performative Post-Scriptum

The title of this post is, of course, a radical reference to core Led Zeppelin track “Trampled Underfoot” (1975).

De Pluribus Unum

A beautifully subtle puzzle:

Scrambled Box Tops

Imagine you have three boxes, one containing two black marbles, one containing two white marbles, and the third, one black and one white marble. The boxes are labelled according to their contents — BB, WW, and BW — but someone has switched the labels so that every box is now incorrectly labelled. You are allowed to take one marble at a time out of any box, without looking inside, and by this process of sampling you are to determine the contents of all three boxes. What is the smallest number of drawings needed to do this? — Martin Gardner, Mathematical Puzzles and Diversions (1959), chapter 3, “Nine Problems”, #5.

Bald eagle, Haliaeetus leucocephalus (Linnaeus 1776)

Bald eagle, Haliaeetus leucocephalus (Linnaeus 1776)

Answer: You can learn the contents of all three boxes by drawing just one marble. The key to the solution is your knowledge that the labels on all three boxes are incorrect. You must draw a marble from the box labelled “black-white”. Assume that the marble drawn is black. You know then that the other marble in the box must be black also, otherwise the label on the box would be correct. Since you have now identified the box containing two black marbles, you can tell at once the contents of the box labelled “white-white”: you know it cannot contain two white marbles, because its label has to be wrong; it cannot contain two black marbles, because you have identified that box; therefore it must contain one black and one white marble. The third box, of course, must then be the one containing two white marbles. You can solve the puzzle by the same reasoning if the marble you draw from the “black-white” box happens to be white instead of black.