Test Your Intuition 55: The Case of Two Screening Tests

Here is a great question invented by Michele Piccione and Ariel Rubinstein. (Let me use this opportunity to recommend their mind boggling 1997 paper on the absent-minded driver.)

The question (TYI 55)

The proportion of newborns with a specific genetic trait is 1%. Two conditionally independent screening tests, A and B, are used to identify this trait in all newborns. However, the tests are not precise. Specifically, it has been found that:

70% of the newborns who are found to be positive according to test A have the trait.
20% of the newborns who are found to be positive according to test B have the trait.

Suppose that a newborn is found to be positive according to both tests. What is your estimate of the probability that this newborn has the trait?

I will try to conduct a poll over Twitter (X). (It looks as it is no longer possible to have a poll over here.) Here is going to be the link to the twitter poll. (Seven days only.)

Michele Piccione and Ariel Rubinstein

Click here for the previous post on Christian Elsholtz, Zach Hunter, Laura Proske, and Lisa Sauermann’s remarkable improvement for Behrend’s construction.

I have a feeling that TYI55 has the potential to be a blockbuster like TYI30 on Mossel’s amazing dice problem.

The Very Short Poll

What is your estimate that the new born has the trait?

(Please give your answer (just the number) as a comment)

The Short Poll

What is your estimate that the new born has the trait

(i) Below 40%

(ii) Between 40% and 80%

(iii) Above 80%

On twitter
Update: here are the outcomes

The Long Poll

What is your estimate that the new born has the trait

(a) 14%

(b) Above 14% but below 20%

(c) 20%

(d) Above 20% and below 45%

(e) 45%

(f) Above 45% below 70%

(g)  70%

(h) Above 70% below 76%

(i) 76%

(j) Above 76% below 94%

(k) 94%

(l) Above 94%

(If twitter will support it I will also post this long poll on twitter. Update: Twitter does not support more than 4 answers.)

_______________________________

Here (a) is the product of the two probabilities ; (c) is the minimum; (e) is the average; (g) is the maximum; and (i) is one minus the product of the complemented probabilities .

_______________________________

After you tested your own intuition, here is the link to Michele and Ariel’s paper. Failing to Correctly Aggregate Signals, Michele Piccione and Ariel Rubinstein  pdf