The Preface to Paul Nahin's new book, "Will You Be Alive 10 Years From Now," includes several wonderful probability problems… I'll likely use them here over time, and will start with the one that seems the most problematic even though on the surface it appears simple.
Columnist Marilyn vos Savant is famous for introducing the 'Monty Hall problem' to the public and giving the correct answer even when many professional mathematicians were initially labeling her "wrong."
Nahin argues that in a different example she WAS wrong, though I think he simply misinterprets matters. The initial, simple question that a reader asked Marilyn in this case was:
“Say you plan to roll a die 20 times. Which result is more likely:
(a) 11111111111111111111 or (b) 66234441536125563152”
Marilyn answered that both sequences were equally likely as outcomes from such a procedure… there is little controversy over that answer (and Nahin agrees with it)… from a strictly frequentist view of rolling a fair die, all sequence-outcomes being equally likely (that likelihood being very small, BTW). BUT, then Marilyn went on to note, “But let’s say you rolled a die out of my view and then said the results were one of those series. Which is more likely? It’s (b) because the roll has already occurred. It was far more likely to have been that mix than a series of ones.”
I don't really have much difficulty with that answer either, but Nahin takes her to task claiming the answer is "wrong" and the probabilities are still equal… that "rolling the die out of view" has no consequence. But clearly there is a difference between anticipating in advance a resultant sequence out of ALL the possible sequences that a procedure might produce, versus addressing just two given sequences AFTER a procedure has already taken place. Nahin faults vos Savant for essentially changing the original question, BUT she clearly states that that is what she is doing (in order to make what I think is an interesting and worthwhile point; it's almost a sort of frequentist vs. Bayesian distinction).
[One way to think about it is simply to make the sequence more ridiculously long: suppose I roll a FAIR die a million times; I record the results and tell you that the outcome was either a million ones, OR, some more-random-looking list of figures… prior to rolling the die both sequences would be equally likely, but with the task already completed, and ONE of the TWO given choices GUARANTEED to be the actual sequence, the second one is more probable.]
Anyway, I find this a good example of how the semantics of a probability problem is often trickier than the math or logic involved (similarly, the precise way the Monty Hall problem is stated and understood is crucial in reaching the right answer under the variety of exact set-ups that can be proposed for it).
[p.s. -- Nahin's Preface offers a second example of vos Savant getting a problem wrong (and later correcting herself), and at some point I'll give that example as well.] [Now, HERE.]
(image credit: Personeoneste/WikimediaCommons)
ADDENDUM: Below are Marilyn's two original responses to this problem in Parade magazine: