By A. M. Yaglom, I. M. Yaglom
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Additional info for Challenging Mathematical Problems With Elementary Solutions, Vol. 1
Next C passes the slip to D after perhaps changing the sign; finally D passes it to an honest judge after perhaps changing the sign. The judge sees a plus sign on the slip. It is known that B, C, and D each change the sign with probability 2/3. What is the probability that A originally wrote a plus? 78a. In certain rural areas of Russia fortunes were once told in the following way. A girl would hold six long blades of grass in her hand with the ends protruding above and below; another girl would tie together the six upper ends in pairs and then tie together the six lower ends in pairs.
Under the hypothesis that the probability of a hit is inversely proportional to the square of the distance, determine the probability that the hunter succeeds in hitting the fox. 77. The problem of the four liars. It is known that each of four people, A, B, C, and D, tells the truth in only one case out of three. Suppose that A makes a statement, and then D says that C says that B says that A was telling the truth. What is the probability that A was actually telling the truth? Remark. This problem can also be formulated in the following way.
Let there be given an infinite sequence of numbers a1, a2, a3, . . Suppose that the first N of these numbers are written on N slips of paper, the slips thoroughly mixed, and then one of them drawn at random. This experiment has N equally probable outcomes; if we denote by q(N) the number of members of the sequence a1, a2, a3, . . , aN which possess some given property, then the probability that the slip drawn bears a number possessing this property is q(N)/N. Suppose that as N→ ∞ the ratio q(N)/N approaches a limit; in this case this limit is called the probability that a number selected at random from the entire sequence has the desired property.