Brain Teasers on Lying Politicians

2. First we ask how probable it is that Alice utters a true statement and Bob makes a true statement of support. Since they both tell the truth 1/4 of the time, these events will both turn out to be true 1/16 of the time (1/4 x 1/4). Now we ask how probable it is that Bob will make a statement of support. Since Bob will utter his support when either both he and Alice tell the truth or when they both lie, the probability of this is 10/16 (1/4 x 1/4 + 3/4 x 3/4). Thus the probability that Alice is telling the truth given that Bob supports her is 1/10 ( the ratio of 1/16 to 10/16). The moral: Confirmation of a very dishonest person's unreliable statement by another very dishonest person makes the statement even less reliable.

3. Consider all possible sequences of 10 statements, no 2 consecutive ones of which are false. Some of these sequences end with a T and some with an F. There are exactly as many 10-element sequences ending in T as there are 9-element sequences of T's and F's, since any sequence of either of these types can be turned into one of the other type by adding or subtracting a T at the end. Furthermore, there are as many 10-element sequences ending in F as there are 8-element sequences of T's and F's, since any sequence of either of these types can be turned into one of the other type by adding or subtracting a TF at the end.

Putting these two facts together shows us that there are just as many 10-element sequences of T's and F's as there 8-element sequences and 9-element sequences put together. In particular, there are as many 3-element sequences as there are 2-element and 1-element sequences combined, as many 4-element sequences as there are 3-element and 2-element sequences combined, as many 5-element sequences as there are 4-element and 3-element sequences combined, and so on. But this is the definition of the famous Fibonacci sequence, each of whose terms after the second is the sum of its two predecessors. So the answer to the question is that there are 144 possible 10-element sequences of T's and F's having no 2 consecutive F's. Note that the "Fib" in "Fibonacci" acquires a new resonance.

Maybe to the polemical books listed above I should add Lying Politicians and the Convoluted Mathematical Truths They Reveal.

Professor of mathematics at Temple University and adjunct professor of journalism at Columbia University, John Allen Paulos is the author of several best-selling books, including Innumeracy, and the just released A Mathematician Plays the Stock Market. His Who’s Counting? column on ABCNEWS.com appears the first weekend of every month.

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