Consider this brain teaser. There are two tables, each with a black can and a grey can. Each can contains a combination of white chips and purple chips, thoroughly mixed. At each table, you will be blindfolded and allowed to draw one chip from your choice of the black can or the grey can. White chips are worthless, but if you draw a purple chip you win \$100.

Table A has this combination of chips:

• Black can: 5 purple chips, 6 white chips
• Grey can: 3 purple chips, 4 white chips

You note that 5/11 is better odds than 3/7, so you choose to draw from the black can at Table A.

Table B has this combination:

• Black can: 6 purple chips, 3 white chips
• Grey can: 9 purple chips, 5 white chips

Since 6/9 is better odds than 9/14, you decide to draw from the black can at Table B as well.

## Bonus Round

Since you’ve been doing so well, you get another chance to try to draw a purple chip. This time, we simply combine the two tables into one. All the chips from the two black cans go into a single black can, and all the chips from the two grey cans go into a single grey can. With this combination, which can will you choose to draw from?

If the black can beats the grey can on Table A, and the black can beats the grey can on Table B, it certainly seems like the combined black cans should beat the combined grey cans, but that is not the case.

Let’s do the math for the combined cans:

• Combined black can: 11 purple chips, 9 white chips
• Combined grey can: 12 purple chips, 9 white chips

Drawing from the black can gives us odds of 11/20, while the odds for the grey can are 12/21, so the grey can is now the better choice!

This version of Simpson’s paradox was popularized by Martin Gardner. The effect was first described by Edward H. Simpson in 1951. In a real-life example of this statistical paradox, two treatments for kidney stones were tested. A 1986 study reported that Treatment A was more successful than Treatment B on large stones, and Treatment A was also more successful on small stones, but when the data were combined, Treatment B was more successful overall.