Chance News 68

Paulos on probability

“Stories vs. Statistics”
by John Allen Paulos, The New York Times, October 24, 2010

In this article, Paulos discusses two classic probability problems, and 143 bloggers[1]respond en masse.

PROBLEM 1. With respect to the story of Linda the famous feminist bank teller, Paulos says that, while more details about a fictional character may make the story more believable, “the more details there are in a story, the less likely it is that the conjunction of all of them is true.”

In the classic illustration of the fallacy …, a woman named Linda is described. She is single, in her early 30s, outspoken, and exceedingly smart. A philosophy major in college, she has devoted herself to issues such as nuclear non-proliferation. So which of the following is more likely?

a.) Linda is a bank teller.

b.) Linda is a bank teller and is active in the feminist movement.

Paulos concludes, “Although most people choose b.), this option is less likely since two conditions must be met in order for it to be satisfied, whereas only one of them is required for option a.) to be satisfied.”

PROBLEM 1. Bloggers
No. 10. a.) is not necessarily more likely than b.). Paulos’ “understanding of others is weak,” and we need more information about how her activism may have led to sexual discrimination to answer the question.
No. 20’s response to #10. “Unless you believe that there could not possibly be such a person, Paulos is correct.”

PROBLEM 2, with two variations. With respect to the classic two-boy problem, Paulos says that “our judgment of [a] probability is almost always affected by its intensional [sic] context.”

Given that a family has two children and that at least one of them is a boy, what is the probability that both children are boys? The most common solution notes that there are four equally likely possibilities — BB, BG, GB, GG …. Since we’re told that the family has at least one boy, the GG possibility is eliminated and only one of the remaining three equally likely possibilities is a family with two boys. Thus the probability of two boys in the family is 1/3. …. What if instead of being told that the family has at least one boy, we meet the parents who introduce us to their son? Then there are only two equally like possibilities — the other child is a girl or the other child is a boy, and so the probability of two boys is 1/2.

Paulos poses a “new variant of the two-boy problem”:

A couple has two children and we’re told that at least one of them is a boy born on a Tuesday. What is the probability the couple has two boys? Believe it or not, the Tuesday is important, and the answer is 13/27. If we discover the Tuesday birth in slightly different intensional [sic] contexts, however, the answer could be 1/3 or 1/2.

PROBLEM 2, with variations. Bloggers
No. 5. “[B]irth order is not part of the question, only whether the unidentified child is male or female, in which case the odds are 1 in 2 ….”
No. 7. Settembrini gives the calculations for the 13/27 problem.
No. 25, Ivan of NYC. “There is another version to the BG problem: suppose that in a family of two children, one is a girl named Emily. What is the probability that the other child is her sister? Turns out to be 1/2. (I was asked this question on an interview)”

Discussion

1. How would you respond to Blogger 10?
2. Suppose that you believe that there could not possibly be such a person as Linda. What would you say is the probability of a.)? The probability of b.)? Would Paulos be correct in this case?
3. How would you respond to Blogger 5?
4. Without viewing Blogger 7’s calculations, calculate the answer to the 13/27 problem.
5. Do you agree with Blogger 25?

Submitted by Margaret Cibes