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A stopping problem

Henk Tijms: Dropping balls Into bins
By Gary Antonick, "Numberplay" blog, New York Times, 9 February 2015

Here is the puzzle, as posed by Prof Tijms:

A game machine is used to drop balls into four bins. The balls are dropped one at a time and any ball will land at random into one of the bins. You can turn off the machine whenever you wish. At the end of the game you win a dollar for every bin containing exactly one ball and you lose half a dollar for every bin containing two or more balls. What stopping rule will maximize your expected gain? In other words, when should you turn off the machine?

He notes that B=4 bins is the first case that gets very complicated, and asks for a heuristic for the general case.

Later he cites a careful reader's correction of his initial solution in the B=2 case: the optimal strategy is to continue until each bin has at least one ball, for an expected winning of 1.25. With B=3,

Thanks to Henk Tijms for sending this link, along another balls and urns story, this one about drawings for the the schedule of European soccer tournaments. He adds that many fans are suspicious of the governing organization FIFA (Fédération Internationale de Football Association).

Was the UEFA Champions League draw rigged?—Bayesian analysis by Henk Tijms
William M. Briggs statistics blog, 5 April 2013

For an accessible general introduction to the theory of optimal stopping see:

Knowing when to stop
by Theodore Hill, American Scientist, March-April 2009

Hill's discussion includes variations on the classical Secretary (a.k.a. Marriage or Dowry) Problem, the Chow-Robbins game, and more.

Submitted by Bill Peterson

Some math doodles

[math]P \left({A_1 \cup A_2}\right) = P\left({A_1}\right) + P\left({A_2}\right) -P \left({A_1 \cap A_2}\right)[/math]


Ask Well: Does kinesiology tape really work?
by Gretchen Reynolds, "Well" blog, New York Times, 27 March 2015

The technical paper referred to is

Kinesiology tape does not facilitate muscle performance: A deceptive controlled trial
by K.Y. Poon,, Manual Therapy, February 2015 (Vol 20, Issue 1, pp. 130–133)

The sample size started at 46, eventually 30 completed the study; each was blindfolded so that the subjects could not see what kind of taping was done: "Thirty healthy participants performed isokinetic testing of three taping conditions: true facilitative KinTape, sham KinTape, and no KinTape." Here are the ANOVA results for Normalized Peak Torque (NPT), Normalized Total Work (NTW), and Time to Peak Torque (TPT):

All the participants were confirmed to be ignorant about KinTape at the debriefing after the experiment. None of them used KinTape prior to the study and they had never heard of the application of KinTape in any circumstances. NPT, NTW, and TPT in different conditions were shown in Table 1. There was no significant difference in NPT between all three taping conditions at 60° (F(2,87) = 0.05, p = 0.96) and 180°/s (F(2,87) = 0.41, p = 0.66). Similar results were found in NTW (F(2,87) = 0.27, p = 0.76; F(2,87) = 0.53, p = 0.59) and TPT (F(2,87) = 0.03, p = 0.98; F(2,87) = 0.32, p = 0.73) at slow and fast contraction speed respectively.

With such enormously high p-values (correspondingly low F-values) the conclusion is

The present study demonstrated that the KinTape application did not generate higher peak torque, yield greater total work, or shorten time to peak torque in healthy young adults. Positive results in the previous studies of KinTape may be attributed to the placebo effects.

Submitted by Paul Alper

Accidental insights

My collective understanding of Power Laws would fit beneath the shallow end of the long tail. Curiosity, however, easily fills the fat end. I long have been intrigued by the concept and the surprisingly common appearance of power laws in varied natural, social and organizational dynamics. But, am I just seeing a statistical novelty or is there meaning and utility in Power Law relationships? Here’s a case in point.

While carrying a pair of 10 lb. hand weights one, by chance, slipped from my grasp and fell onto a piece of ceramic tile I had left on the carpeted floor. The fractured tile was inconsequential, meant for the trash.


As I stared, slightly annoyed, at the mess, a favorite maxim of the Greek philosopher, Epictetus, came to mind: “On the occasion of every accident that befalls you, turn to yourself and ask what power you have to put it to use.” Could this array of large and small polygons form a Power Law? With curiosity piqued, I collected all the fragments and measured the area of each piece.

Piece Sq. Inches  % of Total
1 43.25 31.9%
2 35.25 26.0%
3 23.25 17.2%
4 14.10 10.4%
5 7.10 5.2%
6 4.70 3.5%
7 3.60 2.7%
8 3.03 2.2%
9 0.66 0.5%
10 0.61 0.5%
Montante plot1.png

The data and plot look like a Power Law distribution. The first plot is an exponential fit of percent total area. The second plot is same data on a log normal format. Clue: Ok, data fits a straight line. I found myself again in the shallow end of the knowledge curve. Does the data reflect a Power Law or something else, and if it does what does it reflect? What insights can I gain from this accident? Favorite maxims of Epictetus and Pasteur echoed in my head: “On the occasion of every accident that befalls you, remember to turn to yourself and inquire what power you have to turn it to use” and “Chance favors only the prepared mind.”

Montante plot2.png

My “prepared” mind searched for answers, leading me down varied learning paths. Tapping the power of networks, I dropped a note to Chance News editor Bill Peterson. His quick web search surfaced a story from Nature News on research by Hans Herrmann, et. al. Shattered eggs reveal secrets of explosions. As described there, researchers have found power-law relationships for the fragments produced by shattering a pane of glass or breaking a solid object, such as a stone. Seems there is a science underpinning how things break and explode; potentially useful in Forensic reconstructions. Bill also provided a link to a vignette from CRAN describing a maximum likelihood procedure for fitting a Power Law relationship. I am now learning my way through that.

Submitted by William Montante

The p-value ban